Problem 48 - University Physics with Modern Physics

Question

At room temperature, identical gas cylinders contain 10 moles of nitrogen gas and argon gas, respectively. Determine the ratio of energies stored in the two systems. Assume ideal gas behavior.

Step-by-Step Solution

Answer

Answer: The ratio of energies stored in the Nitrogen and Argon gas cylinders is 5/3 (approximately 1.67).

Step 1: Recall the Ideal Gas Internal Energy formula

To find the energy stored in an ideal gas, we can use the formula for Internal Energy (U) of an ideal gas system: U = n * C_v * T where n is the number of moles, C_v is the molar specific heat at constant volume, and T is the temperature in Kelvin.

Step 2: Define the given values

We are given the following information: - Each cylinder contains 10 moles of gas (n = 10 moles). - Nitrogen (N2) and Argon (Ar) gases are diatomic and monatomic, respectively. - Room temperature is assumed to be 298 K.

Step 3: Find the molar specific heat for Nitrogen and Argon gases

For a diatomic ideal gas (such as Nitrogen, N2), the molar specific heat at constant volume (C_v) is given by: C_v_diatomic = \(\frac{5}{2}R\) For a monatomic ideal gas (such as Argon, Ar), the molar specific heat at constant volume (C_v) is given by: C_v_monatomic = \(\frac{3}{2}R\) Where R is the Universal gas constant, approximately 8.314 J/mol*K.

Step 4: Calculate the internal energy of Nitrogen and Argon gases

Now, we can find the internal energy stored in each cylinder. Using the formula U = n * C_v * T and substituting the given values, we get: For Nitrogen (N2): U_N2 = n * C_v_diatomic * T = 10 * \(\frac{5}{2}R\) * 298 For Argon (Ar): U_Ar = n * C_v_monatomic * T = 10 * \(\frac{3}{2}R\) * 298 Now simplify: U_N2 = 10 * \(\frac{5}{2}(8.314)\) * 298 U_Ar = 10 * \(\frac{3}{2}(8.314)\) * 298

Step 5: Find the energy ratio

Finally, to find the ratio of energies stored in the Nitrogen and Argon gas cylinders, divide U_N2 by U_Ar: Energy Ratio = \(\frac{U_{N_{2}}}{U_{Ar}}\) = \(\frac{10 * \frac{5}{2}(8.314) * 298}{10 * \frac{3}{2}(8.314) * 298}\) Since the factors 10, 8.314, and 298 appear in both numerator and denominator, we can cancel them out: Energy Ratio = \(\frac{\frac{5}{2}}{\frac{3}{2}}\) Now simplify: Energy Ratio = \(\frac{5}{3}\) So, the ratio of energies stored in Nitrogen and Argon gas cylinders is \(\frac{5}{3}\) (approximately 1.67).