A gas container is thermally isolated except for a small hole that insures that the pressure inside the container is equal to the atmospheric pressure p0. Initially, the container holds Ni moles of gas at a temperature Ti. The molar specific heat of the gas at constant pressure is cp. The gas is

Question

A gas container is thermally isolated except for a small hole that insures that the pressure inside the container is equal to the atmospheric pressure [latex]p_0[/latex]. Initially, the container holds [latex]N_i[/latex] moles of gas at a temperature [latex]T_i[/latex]. The molar specific heat of the gas at constant pressure is [latex]c_p[/latex]. The gas is heated up to a temperature [latex]T_f[/latex] by a resistive coil in the cylinder. As the gas temperature rises, some of the gas is released through the small hole. Assume that for the gas remaining in the cylinder, the process is reversible and neglect the specific heat of the heater. Determine:
a) the volume [latex]V_0[/latex] of the container.
b) the number of moles ΔN leaving the container in this process.
c) the heat transfer [latex]Q_{if}[/latex] to accomplish this process.

Numerical Application:

[latex]p_0 = 10^5 Pa, N_0 = 10 moles, T_0 = 273 K, c_p = 29.1 J K^{−1} mol^{−1}, T_f = 293 K[/latex].

Accepted Answer

a) The volume of the container is,

[latex] V_0 =\frac{N_iR T_i}{p_0}= 227 l [/latex].

b) The final number of moles is,

[latex] N_f = \frac{p_0 V_0}{R T_f} = N_i \frac{T_i}{ T_f}[/latex].

Thus, the number of moles leaving the container in this process is,

[latex] ΔN = N_i − N_f = N_i \biggl(1-\frac{T_i}{T_f}\biggr) = 0.68[/latex] mol.

c) Using the result established in b) the heat transfer yields,

[latex] Q_{if} = \int_{i}^{f}{N c_p dT} = c_p N_i T_i \int_{T_i}^{T_f}{\frac{dT}{T} } = c_p N_i T_i \ln \biggl(\frac{T_f}{T_i}\biggr) = 5.6 kJ [/latex].

Question 5.3: A gas container is thermally isolated except for a small hol...

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