Work done and Heat Transferred
Applying the first law of thermodynamics to the process
dU = dQ - dW
Replacing dW with the reversible work
dU = dQ - PdV
since the volume is constant dV = 0 and
dU = dQ
using the definition of the specific heat at constant volume
to replace dU in the first law
For a constant volume process, the addition or removal of heat will lead to a change in the temperature and pressure of the gas, as shown on the two graphs above
Entropy Change
To find the Entropy change, start with the expression derived from the first law
dU = dQ
and replace dU using the definition of specific heat at constant volume and dQ using the definition of entropy
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Work done and Heat Transferred
Applying the first law of thermodynamics to the process
dU = dQ - dW
Replacing dW with the reversible work
dU = dQ - PdV
The volume will change as the gas is heated at constant pressure. To make calculations more straight forward, use ENTHALPY, H
H = U + PV
dH = dU + PdV + VdP
rearrange for dU
dU = dH - PdV - VdP
and substitute into the first law
dH - PdV - VdP = dQ - PdV
the PdV terms cancel out and since pressure is constant dP = 0, so that
dH = dQ
The definition of the specific heat at constant pressure
is used to replace dH in the first law
During a constant pressure process, heat is added or removed and the temperature and volume change. The volume at the end of the process can be found using the ideal gas law and the work calculated from
Entropy Change
Starting with the first law expression for the process
dH = dQ
and replacing dH from the definition of specific heat at constant pressure and Dq from the definition of entropy
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Isothermal Expansion and Compression
Work Done and Heat Transferred
In an isothermal process, the temperature is constant. Applying the first law of thermodynamics to this closed process
For an ideal gas, the internal energy is a function of temperature only, and since the temperature is constant, then dU is zero and
dQ = dW = PdV
using the ideal gas law and integrating between the start and end of the process
This equation tells us that if we do some work on a gas to compress it, the same amount of energy will appear as heat transferred from the gas as it is compressed.
Entropy Change
The Entropy change comes from the equation which incorparates the first and second laws. The energy balance is the first law, and the heat transfer is expressed as an entropy change which is a statement of the second law.
dU = TdS - PdV
dU is zero because the process is isothermal and the working fluid is an ideal gas, so that
TdS = PdV
substituting for the pressure from the ideal gas law for the pressure
and finally integrating between the start and end of the process
Isothermal compression is shown above on P-V and T-S diagrams. Note that as the gas is compressed heat is given out and that as it expands heat is absorbed.
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Work done and Heat Transferred
Applying the first law of thermodynamics to the process
dU = dQ - dW
since no heat is transferred dQ=0 and
dU = - dW
replacing dW with the reversible work term and dU from the definition of specific heat at constant volume
is replaced using the relationship between the specific heat at constant pressure and the specific heat at constant volume, for an ideal gas
This is done because the ratio of specific heats does not vary with temperature.
integrating between the start and end of the process
and
or using the ideal gas law to replace V 
These equations relate P,V and T at the start of the process to P,V and T at the end of the process. When the temperatures at the start and end of the process are known, the work done is calculated from
For an adiabatic process, the work done on or by the gas causes the temperature, pressure and volume to vary as shown in the graphs above
Entropy Change
There is no heat transfer to or from the gas and the process is reversible so that
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