Thermo: Dependence of pressure on entropy at a constant temperature

In summary, the Maxwell relation can be used to prove that (dP/dS)T = κPV at a constant temperature. By substituting the relation κ = α/V, the solution -1/(α*V) can be written as -1/(κPV), which is the desired result.
  • #1
anwat21
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Homework Statement


I was asked to prove that (dP/dS)T (subscript T ie, at a constant temperature) equals κPV ("kappa"PV, or, isothermal compressibility x pressure x volume).
By using the Maxwell relation -(dS/dP)T = (dT/dV)P I got an answer of -1/(alpha*volume) but cannot find out how to make this into kPV.. unless the question is wrong in the first place.



Homework Equations



Maxwell relation -(dS/dP)T = (dT/dV)P

The Attempt at a Solution



By using the Maxwell relation -(dS/dP)T = (dT/dV)P I got an answer of -1/(α*V) but cannot find out how to make this into κPV.. unless the question is wrong in the first place. Ty.
 
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  • #2


Hi there, it seems like you are on the right track! The Maxwell relation you used relates the isothermal compressibility (κ) to the thermal expansion coefficient (α) through the relation κ = α/V. So, substituting this into your answer, you get -1/(α*V) = -1/[(κV)*V] = -1/(κPV), which is the desired result. Therefore, your answer is correct and the question is not wrong. Keep up the good work!
 

1. What is the relationship between pressure and entropy at a constant temperature?

The relationship between pressure and entropy at a constant temperature is described by the equation: P = T(S2 - S1)/(V2 - V1), where P is pressure, T is temperature, S2 and S1 are the final and initial entropy values, and V2 and V1 are the final and initial volumes. This equation is known as the Clausius-Clapeyron equation.

2. How does pressure affect entropy at a constant temperature?

Pressure and entropy have an inverse relationship at a constant temperature. This means that as pressure increases, entropy decreases, and vice versa. This relationship is due to the fact that an increase in pressure leads to a decrease in volume, which results in a decrease in the number of microstates available to the system, therefore decreasing its entropy.

3. Why is it important to study the dependence of pressure on entropy at a constant temperature?

Studying the dependence of pressure on entropy at a constant temperature is important because it helps us understand the behavior of gases and liquids under different conditions. This knowledge is crucial in fields such as thermodynamics, chemical engineering, and atmospheric science, where pressure and temperature play a significant role in the behavior of substances.

4. How does the dependence of pressure on entropy at a constant temperature relate to the second law of thermodynamics?

The dependence of pressure on entropy at a constant temperature is directly related to the second law of thermodynamics, which states that the total entropy of a closed system will always increase over time. This means that as pressure increases, entropy decreases, thus maintaining the overall increase in entropy for the system.

5. What are some real-life applications of the dependence of pressure on entropy at a constant temperature?

The dependence of pressure on entropy at a constant temperature has various real-life applications, such as predicting and understanding phase transitions in substances, calculating the boiling and freezing points of liquids, and designing efficient refrigeration and air conditioning systems. It also plays a crucial role in the study of atmospheric phenomena, such as weather patterns and cloud formation.

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