Question
Asked Nov 8, 2019
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Derive a formula for the entropy of change in a gas that is absorbing heat causing it to go from Tto TH. Express in terms of specific heat of gas, number of moles, R, and TL, TH.

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Expert Answer

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Step 1
Assume that the gas is ideal and the pressure of the gas is constant. The equation for the change
in entropy is given by
do
ds
т
Here, ds is the change in entropy, dQ is the change in heat and T is the temperature
Rewrite the above equation for dQ
dQ Tds ()
From first law of thermodynamics,
dU dQ-pdV
Here, dU is the change in internal energy of the gas, p is the pressure and dV is the change in
volume of the gas.
Put equation (1) in the above equation
dU Tds-pdV (2)
The equation for the specific heat capacity at constant volume is given by
ndU
dT
Here, C is the specific heat capacity at constant volume, n is the number of moles of the gas
and dT is the change in temperature
help_outline

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Assume that the gas is ideal and the pressure of the gas is constant. The equation for the change in entropy is given by do ds т Here, ds is the change in entropy, dQ is the change in heat and T is the temperature Rewrite the above equation for dQ dQ Tds () From first law of thermodynamics, dU dQ-pdV Here, dU is the change in internal energy of the gas, p is the pressure and dV is the change in volume of the gas. Put equation (1) in the above equation dU Tds-pdV (2) The equation for the specific heat capacity at constant volume is given by ndU dT Here, C is the specific heat capacity at constant volume, n is the number of moles of the gas and dT is the change in temperature

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Step 2
Rewrite the above equation for dU
dU nCdT
Put the above equation in equation (2)
nCydT Tds - pdV
Tds nCpdT pdV
dT Pav
ds nCT
(3)
т т
The ideal gas equation is given by
pV nRT
Rewrite the above equation,
р nR
т и
Put the above equation in equation (3)
dт
dV
nR
т
ds nC
V
help_outline

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Rewrite the above equation for dU dU nCdT Put the above equation in equation (2) nCydT Tds - pdV Tds nCpdT pdV dT Pav ds nCT (3) т т The ideal gas equation is given by pV nRT Rewrite the above equation, р nR т и Put the above equation in equation (3) dт dV nR т ds nC V

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