A monoatomic ideal gas is sealed within a container featuring a piston fixed in place at one end. How will the heat capacity C of the gas change if the piston is allowed to move freely?

Question

A monoatomic ideal gas is sealed within a container featuring a piston fixed in place at one end.  How will the heat capacity C of the gas change if the piston is allowed to move freely?

Quizplus · Accepted Answer

11ecba2d_131c_9ebd_a3bf_4b3bfdf5f74f_MD0008_00 The molar heat capacity (C) is the amount of heat required to raise the temperature of 1 mole of a substance by 1 degree Kelvin. As such, the heat (Q) associated with n moles of a substance undergoing a temperature change (ΔT) is given by: 11ecba2d_131c_9ebe_a3bf_5b58047af5d4_MD0008_00 Because the internal energy (U) of a monoatomic ideal gas depends on the gas temperature, the molar heat capacity at constant volume (C_V) is not equal to the molar heat capacity at constant pressure (C_P). At constant volume (V), the change in internal energy (ΔU) of the gas is equal to the heat energy: 11ecba2d_131c_c5cf_a3bf_45ae166653b2_MD0008_00 However, at constant pressure (P), ΔU depends on both heat and work (W), because the ideal gas law states that a temperature change from a heat transfer must produce a change in volume (ΔV) under constant pressure: 11ecba2d_131c_c5d0_a3bf_c166c04e7a7a_MD0008_00 11ecba2d_131c_c5d1_a3bf_e3b7cd53e7cd_MD0008_00 If the internal energy change (ie, temperature change) is assumed to be constant between constant volume and constant pressure conditions, the expression for ΔU_V can be substituted into the equation above to give: 11ecba2d_131c_c5d2_a3bf_c30497f0f2eb_MD0008_00 11ecba2d_131c_c5d3_a3bf_9bb3187ed67d_MD0008_00 Moreover, because the ideal gas constant R is equal to 11ecba2d_131c_c5d4_a3bf_23ee850a5abf_MD0008_00 , the equation simplifies to: 11ecba2d_131c_ece5_a3bf_3d7a1cd5c141_MD0008_00 This expression shows that C_V is less than C_P because some heat energy transferred into a gas stored at constant pressure may be converted to pressure-volume work during the expansion or compression of the gas. Therefore, allowing a previously fixed piston to move freely (to maintain constant pressure) will increase the molar heat capacity of a monoatomic ideal gas. (Choices B, C, and D) Heat may transfer into or out of the gas and change the temperature and internal energy of the gas whether the piston is allowed to move or not. However, once the piston becomes mobile, work can occur. As a result, the heat capacity cannot remain constant but will increase. Educational objective: The molar heat capacity of a gas is the amount of heat required to raise the temperature of 1 mole of the gas by 1 degree Kelvin. The molar heat capacity of a gas at constant pressure exceeds the molar heat capacity of a gas at constant volume.

A Monoatomic Ideal Gas Is Sealed Within a Container Featuring