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I'm having a hard time understanding how it is that the change in entropy is lower at high temperatures because of ∆S=q/T, but that an increase in temperature causes an increase in entropy because of ∆S=nCln(T2/T1). Can someone explain the concept of the relationship between temperature and ∆S?
The first equation is at a constant temperature, while the second equation is with changing temperatures. Because of this difference, you should not expect to see the same trend in the entropy. In the second equation, the value from dividing the final temperature by the initial temperature will always be significantly lower value than the constant temperature used in the first equation. This is why the entropy will increase with the second equation but decrease with the first.
The first equation tells you that adding the same amount of heat at a higher constant temperature results in a smaller increase in temperature, but addition of heat still causes increase in entropy. The second equation is for entropy change when the temperature changes as well; this equation tells us that a greater ratio of increase in temperature results in a greater increase in entropy
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