MCAT Physics & Math (CH10) Study Guide

MCAT Physics & Math (CH10) Study Guide AAMC Content Categories: 4B: Importance of fluids for the circulation of blood, gas movement, and gas exchange. 4C: Electrochemistry and electrical circuits. 4E: Atoms, nuclear decay, electronic structure, and atomic chemical behavior. 5A: Unique nature of water and its solutions. 5B: Nature of molecules and intermolecular interactions. This guide focuses on the physics and math principles underlying these biological and chemical concepts. Unit 1: Fluids and Solids 1. What is the definition of density (ρ)? ANSWER: Density is mass per unit volume. ρ = m/V. Its SI unit is kg/m³. 2. What is the definition of pressure (P)? ANSWER: Pressure is force per unit area. P = F/A. Its SI unit is the Pascal (Pa), which is N/m². 3. How does pressure change with depth in a static fluid? ANSWER: Pressure increases linearly with depth due to the weight of the fluid above. The formula is P = P₀ + ρgh, where P₀ is the external pressure, ρ is density, g is gravity, and h is depth. 4. What is the Pascal's Principle? ANSWER: A pressure change applied to an enclosed, incompressible fluid is transmitted undiminished to every portion of the fluid and the walls of its container. 5. What is the Archimedes' Principle? ANSWER: The buoyant force on a submerged or partially submerged object is equal to the weight of the fluid that the object displaces. F_buoy = ρ_fluid * V_displaced * g. 6. An object with a density of 800 kg/m³ is placed in water (ρ = 1000 kg/m³). What will happen? ANSWER: It will float. Since its density is less than the density of the fluid, the buoyant force will be greater than its weight. 7. What is the definition of viscosity? ANSWER: Viscosity is a fluid's internal resistance to flow. It is a measure of its "thickness" or resistance to shear stress. 8. What are the assumptions of an Ideal Fluid for the continuity and Bernoulli equations? ANSWER: Incompressible (constant density), non-viscous (no internal friction), and laminar (smooth, non-turbulent) flow. 9. What does the Continuity Equation (A₁v₁ = A₂v₂) describe? ANSWER: The conservation of mass for a flowing fluid. It states that the flow rate (A*v) is constant. Where cross-sectional area decreases, flow velocity increases. 10. What does Bernoulli's Equation (P + ½ρv² + ρgh = constant) describe? ANSWER: The conservation of energy for a flowing fluid. It relates pressure, kinetic energy per unit volume (½ρv²), and potential energy per unit volume (ρgh). 11. According to Bernoulli's Equation, if the flow velocity of a fluid increases, what happens to the pressure it exerts? ANSWER: The pressure decreases. This inverse relationship is known as the Venturi effect. 12. What is the difference between laminar and turbulent flow? ANSWER: Laminar flow is smooth, orderly flow in parallel layers. Turbulent flow is chaotic, irregular flow with eddies and vortices. 13. What is surface tension? ANSWER: Surface tension is the tendency of a liquid surface to minimize its area, behaving like a stretched elastic membrane. It is caused by cohesive forces between molecules. 14. What is Young's Modulus? ANSWER: Young's Modulus is a measure of a solid material's stiffness or resistance to elastic deformation under tension or compression. It is stress divided by strain (Y = (F/A) / (ΔL/L)). Unit 2: Electrostatics and Circuits 15. What is Coulomb's Law? ANSWER: It gives the magnitude of the electrostatic force between two point charges: F = k|q₁q₂|/r², where k is Coulomb's constant, q are the charges, and r is the distance between them. Like charges repel, opposites attract. 16. How does the electric force change if the distance between two charges is doubled? ANSWER: The force is reduced to one-quarter of its original value, as force is inversely proportional to the square of the distance (1/r²). 17. What is an electric field (E)? ANSWER: A region of space around a charged object where a force would be exerted on another charged object. E = F/q. Its direction is defined as the direction a positive test charge would move. 18. What is electric potential (V), commonly called voltage? ANSWER: Electric potential is the electric potential energy per unit charge. V = U/q. Its SI unit is the Volt (J/C). 19. What is the relationship between electric potential energy (U) and work (W)? ANSWER: The work done by a conservative electric field is equal to the negative of the change in potential energy. W = -ΔU. 20. What is the function of a capacitor? ANSWER: A capacitor stores electric potential energy by storing separated charge. It consists of two conductors separated by an insulator. 21. What is capacitance (C)? ANSWER: Capacitance is the ability of a capacitor to store charge per unit voltage. C = Q/V. Its SI unit is the Farad (F). 22. What factors affect the capacitance of a parallel-plate capacitor? ANSWER: It is directly proportional to the area (A) of the plates and inversely proportional to the distance (d) between them. C = ε₀A/d, where ε₀ is the permittivity of free space. 23. What is the energy (U) stored in a capacitor? ANSWER: U = ½ QV = ½ CV² = ½ Q²/C. 24. What is a dielectric? ANSWER: An insulating material placed between the plates of a capacitor to increase its capacitance by reducing the electric field within the capacitor. 25. Define current (I). ANSWER: Current is the rate of flow of electric charge. I = ΔQ/Δt. Its SI unit is the Ampere (A), which is C/s. 26. What is the difference between direct current (DC) and alternating current (AC)? ANSWER: DC is current that flows in one direction only. AC is current that periodically reverses direction. 27. What is Ohm's Law? ANSWER: The voltage across a conductor is directly proportional to the current flowing through it: V = IR, where R is the resistance. 28. What is resistance (R)? ANSWER: Resistance is a measure of a material's opposition to the flow of electric current. Its SI unit is the Ohm (Ω). 29. What factors determine the resistance of a wire? ANSWER: R = ρL/A, where ρ is the resistivity of the material, L is the length, and A is the cross-sectional area. 30. What is the equivalent resistance for resistors in series? ANSWER: R_eq = R₁ + R₂ + R₃ + ... The equivalent resistance is always larger than any individual resistor. 31. What is the equivalent resistance for resistors in parallel? ANSWER: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + ... The equivalent resistance is always smaller than the smallest individual resistor. 32. What is Kirchhoff's Junction Rule? ANSWER: The sum of all currents entering a junction must equal the sum of all currents leaving it. This is a statement of conservation of charge. 33. What is Kirchhoff's Loop Rule? ANSWER: The sum of the voltage drops around any closed loop in a circuit must equal zero. This is a statement of conservation of energy. 34. How does a voltmeter need to be connected in a circuit? ANSWER: In parallel with the component whose voltage is being measured. It has a very high internal resistance. 35. How does an ammeter need to be connected in a circuit? ANSWER: In series with the component whose current is being measured. It has a very low internal resistance. 36. What is the power (P) dissipated by a resistor? ANSWER: P = IV = I²R = V²/R. Its SI unit is the Watt (W). Unit 3: Atomic and Nuclear Phenomena 37. What is the photoelectric effect? ANSWER: The emission of electrons from a metal surface when light of a sufficiently high frequency shines upon it. 38. What was the key finding of the photoelectric effect that classical physics couldn't explain? ANSWER: That the kinetic energy of the emitted electrons depends only on the frequency of the light, not its intensity. Intensity only affects the number of electrons. 39. How did Einstein explain the photoelectric effect? ANSWER: By proposing that light consists of discrete packets of energy called photons. The energy of a single photon is E = hf, where h is Planck's constant and f is the frequency. 40. What is the work function (Φ)? ANSWER: The minimum energy required to eject an electron from a metal surface. The maximum kinetic energy of the ejected electron is K_max = hf - Φ. 41. What is the threshold frequency? ANSWER: The minimum frequency of light required to cause the photoelectric effect. f_threshold = Φ / h. 42. What is an atomic absorption spectrum? ANSWER: A spectrum of dark lines against a bright background, produced when a cool, diffuse gas absorbs specific wavelengths of light from a continuous source behind it. 43. What is an atomic emission spectrum? ANSWER: A spectrum of bright lines against a dark background, produced when a hot, dense gas or vapor emits light at specific wavelengths. 44. What do these discrete spectra indicate about atomic structure? ANSWER: That electrons in atoms can only exist at specific, discrete energy levels. The lines correspond to transitions between these levels. 45. What is the energy of a photon emitted when an electron drops from a higher energy level (E_high) to a lower one (E_low)? ANSWER: E_photon = E_high - E_low = hf. 46. What is the Bohr model of the atom? ANSWER: A model where electrons orbit the nucleus in specific, stable energy levels without radiating energy. They can "jump" between levels by absorbing or emitting photons. 47. What is nuclear binding energy? ANSWER: The energy equivalent to the mass defect of a nucleus; the energy that would be required to break a nucleus into its separate protons and neutrons. 48. What is the difference between fusion and fission? ANSWER: Fusion is the combining of light nuclei to form a heavier nucleus. Fission is the splitting of a heavy nucleus into lighter nuclei. Both processes release energy. 49. What is alpha (α) decay? ANSWER: The emission of an alpha particle (a helium-4 nucleus: 2 protons, 2 neutrons) from a radioactive nucleus. It decreases the atomic number by 2 and mass number by 4. 50. What is beta-minus (β⁻) decay? ANSWER: The transformation of a neutron into a proton, with the emission of an electron (beta particle) and an antineutrino. It increases the atomic number by 1. 51. What is beta-plus (β⁺) decay / positron emission? ANSWER: The transformation of a proton into a neutron, with the emission of a positron (anti-electron) and a neutrino. It decreases the atomic number by 1. 52. What is gamma (γ) decay? ANSWER: The emission of a high-energy photon (gamma ray) from an excited nucleus. It does not change the atomic or mass number, only the energy of the nucleus. 53. What is half-life (t_{1/2})? ANSWER: The time required for half of the radioactive nuclei in a sample to decay. 54. What is the key difference between a particle and its antiparticle? ANSWER: They have the same mass but opposite charge (and other opposite quantum numbers like lepton number). 55. What happens when a particle and its antiparticle collide? ANSWER: They annihilate each other, converting their mass into energy in the form of photons (e.g., gamma rays). Unit 4: The Unique Nature of Water and Solutions 56. Why is water considered a polar molecule? ANSWER: The oxygen atom is more electronegative than the hydrogen atoms, creating a permanent dipole with a partial negative charge on the oxygen and partial positive charges on the hydrogens. 57. What type of intermolecular force is primarily responsible for water's high boiling point and surface tension? ANSWER: Hydrogen bonding. 58. Why does ice float on liquid water? ANSWER: Because ice is less dense than liquid water. The hydrogen bonds in ice form a crystalline lattice that holds molecules farther apart than in the liquid state. 59. What is the definition of specific heat (c)? ANSWER: The amount of heat energy required to raise the temperature of 1 gram of a substance by 1 degree Celsius. Water has a very high specific heat (1 cal/g°C). 60. What is the definition of heat of vaporization? ANSWER: The amount of heat energy required to convert 1 gram of a liquid into vapor at its boiling point. Water has a very high heat of vaporization. 61. What is a solution? ANSWER: A homogeneous mixture composed of a solute (the substance dissolved) and a solvent (the substance doing the dissolving, usually present in greater amount). 62. What is the general rule for solubility of ionic compounds in water? ANSWER: "Like dissolves like." Polar water molecules are effective at dissolving other polar molecules and ionic compounds due to electrostatic interactions. 63. What is concentration? ANSWER: A measure of the amount of solute dissolved in a given amount of solvent or solution. Common units include molarity (M = mol/L). 64. What is a colligative property? ANSWER: A property of a solution that depends only on the number of solute particles present, not on their chemical identity (e.g., vapor pressure lowering, boiling point elevation, freezing point depression, osmotic pressure). 65. How does adding a non-volatile solute affect the boiling point of a solvent? ANSWER: It elevates the boiling point. The boiling point elevation is ΔT_b = i * K_b * m, where i is the van't Hoff factor, K_b is the ebullioscopic constant, and m is molality. 66. How does adding a solute affect the freezing point of a solvent? ANSWER: It depresses the freezing point. The freezing point depression is ΔT_f = i * K_f * m. 67. What is osmosis? ANSWER: The net movement of solvent molecules through a semipermeable membrane from a region of lower solute concentration to a region of higher solute concentration. 68. What is osmotic pressure (π)? ANSWER: The pressure that must be applied to a solution to prevent the inward flow of water across a semipermeable membrane. π = iMRT, where M is molarity, R is the gas constant, and T is temperature. Unit 5: Essential Math for the MCAT 69. How do you convert a number to scientific notation? ANSWER: Write it as a number between 1 and 10 multiplied by a power of 10. E.g., 4500 = 4.5 x 10³; 0.032 = 3.2 x 10⁻². 70. How do you multiply two numbers in scientific notation? ANSWER: Multiply the coefficients and add the exponents. (A x 10ᵃ) * (B x 10ᵇ) = (A*B) x 10ᵃ⁺ᵇ. 71. How do you divide two numbers in scientific notation? ANSWER: Divide the coefficients and subtract the exponents. (A x 10ᵃ) / (B x 10ᵇ) = (A/B) x 10ᵃ⁻ᵇ. 72. How do you add or subtract numbers in scientific notation? ANSWER: First, convert them to have the same power of 10. Then add or subtract the coefficients and keep the common exponent. 73. What are the basic rules of logarithms? (log(AB), log(A/B), log(A^B)) ANSWER: log(AB) = log A + log B; log(A/B) = log A - log B; log(A^B) = B log A. 74. What is the pH formula? ANSWER: pH = -log[H₃O⁺], where [H₃O⁺] is the hydronium ion concentration. 75. If [H₃O⁺] = 1 x 10⁻⁵ M, what is the pH? ANSWER: pH = -log(1 x 10⁻⁵) = 5. 76. If the pH is 3, what is the [H₃O⁺]? ANSWER: [H₃O⁺] = 10^(-pH) = 10⁻³ = 0.001 M. 77. What is the formula for the slope of a line? ANSWER: m = (y₂ - y₁) / (x₂ - x₁) = rise / run. 78. What is the formula for a straight line? ANSWER: y = mx + b, where m is the slope and b is the y-intercept. 79. On a graph of velocity vs. time, what does the slope represent? ANSWER: Acceleration. 80. On a graph of velocity vs. time, what does the area under the curve represent? ANSWER: Displacement. 81. On a graph of force vs. distance, what does the area under the curve represent? ANSWER: Work. 82. How do you calculate the area of a circle? ANSWER: A = πr². 83. How do you calculate the circumference of a circle? ANSWER: C = 2πr. 84. How do you calculate the volume of a cylinder? ANSWER: V = πr²h. 85. How do you calculate the volume of a sphere? ANSWER: V = (4/3)πr³. 86. What is the Pythagorean Theorem? ANSWER: For a right triangle with legs a and b and hypotenuse c, a² + b² = c². 87. What are the trigonometric functions in a right triangle? (SOH CAH TOA) ANSWER: sin(θ) = Opposite / Hypotenuse; cos(θ) = Adjacent / Hypotenuse; tan(θ) = Opposite / Adjacent. 88. How do you convert from degrees to radians? ANSWER: Multiply by (π radians / 180°). 89. What is a vector quantity? ANSWER: A quantity that has both magnitude and direction (e.g., force, velocity, displacement). 90. What is a scalar quantity? ANSWER: A quantity that has only magnitude, no direction (e.g., mass, speed, temperature, energy). 91. How do you find the magnitude of a resultant vector from its components (A_x, A_y)? ANSWER: |A| = √(A_x² + A_y²). 92. How do you find the direction (angle) of a resultant vector from its components? ANSWER: θ = tan⁻¹ (A_y / A_x). Unit 6: Passage-Based & Integrated Questions 93. A red blood cell (which contains solutes) is placed in pure water. Describe what happens and why. ANSWER: The cell will swell and may lyse (burst). The pure water is hypotonic relative to the cell's interior. Water moves into the cell via osmosis to equalize solute concentration. 94. A red blood cell is placed in a very concentrated salt solution. Describe what happens. ANSWER: The cell will shrivel (crenate). The salt solution is hypertonic. Water moves out of the cell via osmosis. 95. In the cardiovascular system, where is blood flow velocity the slowest and why? ANSWER: In the capillaries. According to the continuity equation (A₁v₁ = A₂v₂), the total cross-sectional area of the capillaries is enormous, so flow velocity must be very slow to allow for efficient gas and nutrient exchange. 96. A patient has a blockage in an artery. According to fluid dynamics, how does this affect pressure and flow? ANSWER: The blockage reduces the cross-sectional area, increasing the flow velocity of the blood at the site of the blockage (continuity equation). According to Bernoulli's principle, this increased velocity results in a local decrease in pressure on the arterial wall, which can be dangerous. 97. In an ECG, the electrical activity of the heart is measured. What is physically being measured on the skin's surface? ANSWER: Voltage differences (electric potentials) caused by the depolarization and repolarization of cardiac muscle cells. 98. In an X-ray machine, how are the high-energy photons produced? ANSWER: They are produced when high-speed electrons are rapidly decelerated upon striking a metal target (bremsstrahlung radiation) or when they cause inner-shell electron transitions in the target atoms. 99. A radioactive tracer with a half-life of 6 hours is injected into a patient. What fraction of the original sample remains after 18 hours? ANSWER: 18 hours is 3 half-lives (18/6=3). Fraction remaining = (1/2)^3 = 1/8. 100. In gel electrophoresis, why do DNA fragments separate by size? ANSWER: The phosphate backbone of DNA gives it a uniform negative charge per unit length. When placed in an electric field, all fragments experience the same force per charge. However, smaller fragments experience less drag/resistance from the gel matrix and thus migrate faster and farther than larger fragments. 101. The dissociation of a weak acid is given by HA ⇌ H⁺ + A⁻. If the pH of the solution is equal to the pK_a of the acid, what is the relationship between [HA] and [A⁻]? ANSWER: They are equal. Using the Henderson-Hasselbalch equation: pH = pK_a + log([A⁻]/[HA]). If pH = pK_a, then log([A⁻]/[HA]) = 0, so [A⁻]/[HA] = 1. 102. Why does sweating cool the body? ANSWER: It utilizes water's high heat of vaporization. As sweat evaporates, it absorbs a large amount of heat energy from the skin, cooling it down. 103. When calculating the electric force between a proton and an electron in a hydrogen atom, which fundamental force is being described? ANSWER: The electromagnetic force. 104. In nuclear medicine, why are beta-emitters often preferred over alpha-emitters for internal therapies? ANSWER: Beta particles (electrons) have a greater penetration depth than alpha particles but are still short enough to deposit most of their energy within a small, targeted tissue volume, minimizing damage to surrounding healthy tissue. Alpha emitters are too destructive for most internal uses. 105. A graph of reaction rate vs. substrate concentration for an enzyme plateaus at high substrate concentration. What does this saturation indicate? ANSWER: It indicates that all active sites of the enzyme are occupied. The reaction rate becomes limited by the turnover rate of the enzyme (V_max), not by substrate availability. This is a classic example of a system following a logarithmic or asymptotic relationship.