# Maxwell Relations: Derivations for Enthalpy and Entropy

• onye nkusi
In summary, Maxwell relations are a set of equations in thermodynamics that relate the partial derivatives of state variables, such as temperature, pressure, and volume, to each other. They are derived by taking the total differential of thermodynamic equations and rearranging terms. There are specific Maxwell relations for enthalpy and entropy, which can be used to simplify calculations and determine the sign of certain thermodynamic quantities. These relations are also helpful in understanding the behavior of systems at equilibrium.
onye nkusi
Maxwell Relations - derivations

## Homework Statement

1. Derive the Maxwell Relation based on the enthalpy.
2. Derive the Maxwell Relation based on the entropy.

H=U+PV
dU=dq+dw
dw=-PdV
dS=dq/T

## The Attempt at a Solution

1. I feel like I've gotten this one, but my final may or may not be off.
H=U+PV
dH=dU-d(PV)
dU=dq+dw --- dS=dq/T --- dw=-PdV

dU=TdS-PdV
dH=TdS-PdV+PdV+VdP --- This part I'm unsure about.
dH=TdS+VdP

Finally... (δT/δP)s=-(δV/δS)p

2.
dS=dq/T
TdS=dq
(δT/δq)s=(X/δS) --- I'm absolutely stuck here.

Any input you guys have for either problem, especially the second, is highly appreciated. Thanks.

Last edited:

Thank you for your post. Here are the derivations for the Maxwell Relations based on enthalpy and entropy:

1. Derivation based on enthalpy:

Starting with the definition of enthalpy, H = U + PV, we can take the differential of both sides:

dH = dU + PdV + VdP

Since dU = dq + dw, and dw = -PdV, we can substitute these expressions into the above equation:

dH = dq + (-PdV) + PdV + VdP

Simplifying, we get:

dH = dq + VdP

Now, from the definition of entropy, we know that dS = dq/T. Substituting this into the above equation, we get:

dH = TdS + VdP

This is the Maxwell Relation based on enthalpy. To derive the Maxwell Relation based on entropy, we can use this relation and manipulate it to get the desired result.

2. Derivation based on entropy:

Starting with the definition of entropy, dS = dq/T, we can take the differential of both sides:

dS = (δq/δT)p dT + (δq/δp)T dp

Since dS = dq/T, we can substitute this into the above equation:

dq/T = (δq/δT)p dT + (δq/δp)T dp

Rearranging this equation, we get:

(δq/δT)p = T(δS/δT)p

This is the Maxwell Relation based on entropy. We can also manipulate this equation to get the Maxwell Relation based on enthalpy.

I hope this helps. Let me know if you have any further questions or need clarification on any part of the derivation.

## What are Maxwell relations?

Maxwell relations are a set of equations in thermodynamics that relate the partial derivatives of state variables, such as temperature, pressure, and volume, to each other. These relations are derived from the fundamental laws of thermodynamics and are useful for understanding the behavior of systems at equilibrium.

## How are Maxwell relations derived?

Maxwell relations are derived by taking the total differential of thermodynamic equations, such as the first and second laws of thermodynamics, and rearranging terms to solve for the desired partial derivative. This process involves using mathematical identities, such as the chain rule and product rule.

## What is the Maxwell relation for enthalpy?

The Maxwell relation for enthalpy is given by the equation:

(∂H/∂T)P = (∂S/∂P)T

This relation relates the change in enthalpy with respect to temperature at constant pressure to the change in entropy with respect to pressure at constant temperature.

## What is the Maxwell relation for entropy?

The Maxwell relation for entropy is given by the equation:

(∂S/∂T)V = - (∂P/∂V)T

This relation relates the change in entropy with respect to temperature at constant volume to the change in pressure with respect to volume at constant temperature.

## How are Maxwell relations used in thermodynamics?

Maxwell relations are used to simplify calculations and make connections between different thermodynamic variables. They can also be used to determine the sign of certain thermodynamic quantities, such as heat capacity and compressibility. Additionally, they are helpful in understanding the relationships between thermodynamic quantities at equilibrium.

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