How Is Heat Calculated from Entropy and Temperature Changes in Thermodynamics?

[jmcgraw]

I am given the number of moles of an ideal monotomic gas. I am also given a chart with a graph of temperature plotted as a function of entropy.

One of the questions is: find the heat that was transferred to the gas.

How do I find that? I thought about saying:

dS = dQ/T <=> dQ = T dS

then I found the slope of the graph with respect to S and found the y-intercept. So now I had T in terms of S and I said:

dQ = (mS + b) dS

taking the integral of both sides I found Q and was able to easily integrate from initial entropy to final entropy. But I got the answer wrong. I don't want to do it all over again, unless I am sure this is the right approach. Is there an easier way?

This is problem #8 in Chapter 21 of Halliday/resnick/walker 6th edition.

 

[jmcgraw]

O.k. I overcame my sloth and did it again. I got the right answer!

But it seems overly involved. Is there an easier way?

 

[quicknote]

Your approach is correct. The equation dS = dQ/T is known as the Clausius inequality and is a fundamental relationship in thermodynamics. It states that the change in entropy of a system is equal to the heat transferred to the system divided by the temperature at which the transfer occurs. In this case, you are given the number of moles of an ideal monatomic gas, which means you can use the ideal gas law (PV = nRT) to find the temperature of the gas.

To find the heat transferred to the gas, you can use the equation dQ = TdS as you have done. However, you may have made a mistake in calculating the slope of the graph with respect to entropy. Make sure you use the correct units for the entropy and temperature values in order to get the correct slope.

Another approach you could try is to use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat transferred to the system minus the work done by the system. In this case, since the gas is ideal and monatomic, there is no work done by the system (W = 0). Therefore, the change in internal energy is equal to the heat transferred to the gas. You can use the ideal gas law to calculate the change in internal energy and then use that value to find the heat transferred to the gas.

Overall, both approaches should give you the same answer if done correctly. If you are still having trouble, I suggest double-checking your calculations and units to make sure they are correct.