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here is the solution to the problem. a) State TWO assumptions used to derive this equation. Here are two assumptions used in the kinetic theory of gases: • The gas consists of a very large number of identical molecules that are in constant, random motion. • The volume occupied by the gas molecules themselves is negligible compared to the total volume of the container. • Collisions between molecules and with the container walls are perfectly elastic. • There are no intermolecular forces between gas molecules except during collisions. b) Use the above equation to show that c^2 is related to the absolute temperature of the gas by; c^2 = (3RT)/(M) Step 1: Start with the given equation and the definition of density. The given equation is P = (1)/(3) c^2. The density of a gas is its mass (m) divided by its volume (V), so = (m)/(V). Substitute this into the pressure equation: P = (1)/(3) ((m)/(V)) c^2 Rearrange to get PV: PV = (1)/(3) m c^2 Step 2: Express the mass of the gas in terms of moles and molar mass. The total mass of the gas (m) can be written as the number of moles (n) multiplied by the molar mass (M) of the gas: m = nM. Substitute this into the equation from Step 1: PV = (1)/(3) (nM) c^2 Step 3: Use the ideal gas law. The ideal gas law states PV = nRT, where R is the molar gas constant and T is the absolute temperature. Substitute nRT for PV in the equation from Step 2: nRT = (1)/(3) (nM) c^2 Step 4: Simplify and solve for c^2. Cancel n from both sides of the equation: RT = (1)/(3) M c^2 Multiply both sides by 3 and divide by M to isolate c^2: c^2 = (3RT)/(M) This shows the desired relationship. c) Determine the root-mean-square speed of the molecules if the gas is heated to a temperature of 150^. Step 1: Convert initial and final temperatures to Kelvin. Initial temperature T_1 = 30.0^ = 30.0 + 273.15 = 303.15 K. Final temperature T_2 = 150^ = 150 + 273.15 = 423.15 K. Step 2: Relate root-mean-square speed to temperature. From part (b), we have c^2 = (3RT)/(M). The root-mean-square speed (c_rms) is the square root of the mean-square speed: c_rms = sqrt(c^2) = sqrt((3RT)/(M)). Since 3R and M are constants for a given gas, c_rms is directly proportional to the square root of the absolute temperature: c_rms sqrt(T). Therefore, we can write the ratio: c_rms,2c_rms,1 = sqrt((T_2)/(T_1)) Step 3: Calculate the new root-mean-square speed (c_rms,2). Given c_rms,1 = 850 m s^-1. c_rms,2 = c_rms,1 sqrt((T_2)/(T_1)) c_rms,2 = 850 m s^-1 sqrt(423.15 K)303.15 K c_rms,2 = 850 m s^-1 sqrt(1.3958) c_rms,2 = 850 m s^-1 × 1.1814 c_rms,2 ≈ 1004.19 m s^-1 Rounding to three significant figures: c_rms,2 ≈ 1000 m s^-1 Drop the next question.