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Professor Mani discusses an example of a solid in contact with a reservoir at a lower temperature. After the system has reached equilibrium the entropy of the system increases. This is demonstrated by an explicit calculation. Several important remarks are made. In particular how the equilibrium can be achieved by a reversible quasi static process, in which case there will be no change in entropy.
Question:
A solid at temperature \(T_i\) with specific heat \(C=C_p=C_V\) of mass 1 kg is kept in contact with a reservoir kept at lower temperature \(T_0 < T_i\). What is the change in entropy of the solid? Change in entropy of the reservoir? Show that the total entropy increases when the system reaches equilibrium.
Solution:
The temperature of the solid decreases from \(T_i\) to \(T_0\). The reservoir being a very large system, its temperature does not change, it remains constant to \(T_0\), despite having received heat from the body.
We shall consider an infinitesimal change in temperature of the solid from \(T\) to \(T-\Delta T\) compute the change in entropy of the reservoir and the solid and integrate from \(T_i\) to \(T_0\).
- Heat supplied to the reservoir is \(\Delta Q= C \Delta T)\). Therefore, increase in entropy of the reservoir \(= \frac{C \Delta T}{T_0}\)
- The total change in entropy of the reservoir \(=\int_{T_0)}^{T_i} \frac{C dT}{T_0} = \frac{C(T_i-T_0)}{T_0}\)
- The decrease in entropy of the solid \(=C \int_{T_0}^{T_i} \frac{dT}{T}= C\ln \frac{T_i}{T_0}\)
- Thus the total change (increase) in entropy of the solid plus reservoir \(= c[ x- 1+ \ln x]\), where \(x=\frac{T_i}{T_0}\). Since \(x>1\), the change in entropy is seen to be positive. The reader may try plotting a graph of function \(= c[ x- 1+ \ln x]\) and check behaviour for \(x>1\)
Remarks
- If the solid had a lower temperature initially, then again the change in entropy for the solid plus reservoir will be seen to be positive
- Had we changed the temperature of the solid by using a series of reservoirs, differing in temperature by infinitesimal amounts, we could have made the process reversible.
This example is taken from a video recording of Prof. Mani's Lecture from a course given at Chennai Mathematical Institute.
By clicking on play list, the part of the original video can be played where this example is discussed.
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Example of calculation of change in entropy of a solid and a reservoir when the solid at a higher temperature comes to equilibrium with the reservoir at a lower temperature. An explicit calculation demonstrates that the change in entropy is positive, as expected for an irreversible process.
Prof Mani make an observation that there will be no change in entropy if the heat transfer takes place through a series of reserviors with temperatures differing by infinitesimal amounts. He also gives a reference where this explicit calculation has been done.
