# Proving thermodynamic relationship

• goggles31
In summary, the conversation discusses the proof of the equation Cv = (∂U/∂T)v = T(∂S/∂T)v, which can be derived using the Maxwell relations. The conversation also touches on the application of the product rule and the division method in differentiating infinitesimal differential quantities. Finally, the conversation addresses the question of finding dS in terms of dT and dV, which can be done by imposing the condition that dV = 0.
goggles31
Cv = (∂U/∂T)v = T(∂S/∂T)v

I can prove this by using the Maxwell relations, but I have trouble deriving it from the first law.

dU = TdS - pdV
(∂U/∂T)v = T(∂S/∂T)v + ∂S(∂T/∂T)v = T(∂S/∂T)v + ∂S

Is there a problem with my derivation?

goggles31 said:
dU = TdS - pdV
(∂U/∂T)v = T(∂S/∂T)v + ∂S(∂T/∂T)v = T(∂S/∂T)v + ∂S
Could you explain how you got the second term?

Fightfish said:
Could you explain how you got the second term?

I applied to product rule to Tds hence obtaining two terms.

goggles31 said:
I applied to product rule to Tds hence obtaining two terms.
You don't apply the product rule here because you are not actually differentiating per se - you are not finding ##\frac{d}{dT} (dU)##, which doesn't make sense because well, what does it mean to differentiate an infinitesimal differential quantity?

Rather what we are doing is simply "division" - the mathematicians will complain here, but what we are doing can be rigorously justified if we so wish. That is to say, we divide by ##dT## throughout and then impose the condition that ##dV = 0##.

goggles31
If S=S(T,V), what is dS in terms of dT and dV?

Zypheros_Knight

## 1. What is the thermodynamic relationship?

The thermodynamic relationship is a mathematical expression that describes the interdependence between different thermodynamic properties, such as temperature, pressure, volume, and energy.

## 2. Why is it important to prove thermodynamic relationships?

Proving thermodynamic relationships is important because it allows scientists to validate the laws of thermodynamics, which are fundamental principles governing the behavior of matter and energy. It also helps in the development of new technologies and the understanding of complex systems.

## 3. How do scientists prove thermodynamic relationships?

Scientists use various experimental techniques, such as calorimetry, to measure and analyze the changes in thermodynamic properties of a system. They also use mathematical models and simulations to validate the relationships between these properties.

## 4. What are some common thermodynamic relationships?

Some common thermodynamic relationships include the ideal gas law, which describes the relationship between pressure, volume, and temperature of a gas, and the heat capacity formula, which relates the change in temperature to the amount of heat transferred.

## 5. Can thermodynamic relationships be applied to all systems?

Thermodynamic relationships can be applied to most systems, as long as they follow the laws of thermodynamics. However, some systems may have complex interactions and require more advanced mathematical models to accurately describe their thermodynamic behavior.

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