# Total Derivatives and Thermodynamics

## Question 1

What are the following quantities functions of: volume, pressure, temperature and mass? For I am very confused when I should be using $$d$$ or $$\partial$$ for my derivatives in thermodynamics.

For example, If I take the total derivative of internal energy [$$U(S,V)$$] as a function of temperature, I get the following:

$$\frac{d}{dT}$$U(S,V) = $$\frac{ \partial U}{ \partial T} + (\frac{ \partial U}{ \partial S}) \frac{ds}{dT} + (\frac{ \partial U}{ \partial V}) \frac{dV}{dT}$$

If I multiply both sides by $$dT$$ to get my desired infinitesimal, I get:

$$dU(S,V) = \frac{ \partial U}{ \partial t} dT + (\frac{ \partial U}{ \partial S}) ds + (\frac{ \partial U}{ \partial V})dV$$

However, my textbook uses the following notation in some questions and I do not think $$\partial t$$ and $$dt$$ cancel each other out, but I am not sure

$$\frac{d}{dT}$$U(S,V) = $$\frac{ \partial U}{ \partial T} + (\frac{ \partial U}{ \partial S}) \frac{ \partial s}{ \partial T} + (\frac{ \partial U}{ \partial V}) \frac{ \partial V}{ \partial T}$$

## Question 2

When you turn to infinitesimal form, which variable are we taking the derivative with respect to? I always assumed time, but my previous question is casting doubles. For example:

$$\frac{d}{dt}$$U(S,V) = $$\frac{ \partial U}{ \partial t} + (\frac{ \partial U}{ \partial S}) \frac{ds}{dt} + (\frac{ \partial U}{ \partial V}) \frac{dV}{dt}$$

Some authors "cheat" and do the derivative calculations without the numerator. For example:

$$F + \Delta F = U + \Delta U - (P + \Delta P)(V + \Delta V)$$

How is this connected to a normal calculus (i.e. total/partial derivatives)?

## Summary

I understand kinds types of derivatives (partial/total), but do know know which type thermodynamics uses or when

## Previous Research

Partial derivatives vs total derivatives in thermodynamics

Thermodynamics is a minefield for issues like this.

The internal energy, as you say, is a function of two variables - $$S$$ and $$V$$. We can define its partial derivatives with respect to $$S$$ and $$V$$ as follows:

$$\left(\frac{\partial U}{\partial S}\right)_V = \lim_{h\rightarrow 0} \frac{U(S+h,V)-U(S,V)}{h}$$

$$\left(\frac{\partial U}{\partial V}\right)_S = \lim_{h\rightarrow 0} \frac{U(S,V+h)-U(S,V)}{h}$$

But now you want to talk about $$\frac{\partial U}{\partial T}$$, where $$T \equiv \left(\frac{\partial U}{\partial S}\right)_V$$, and it's not immediately clear what that means, since $$U$$ is not even a function of $$T$$.

What we mean is the following: we change $$U$$ by changing both $$S$$ and $$V$$ at the same time:

$$dU = U\big(S+dS,V+dV\big) = \left(\frac{\partial U}{\partial S}\right)_V dS + \left(\frac{\partial U}{\partial V}\right)_S dV$$

Because $$T\equiv \left(\frac{\partial U}{\partial S}\right)_V$$ is also a function of $$S$$ and $$V$$, by changing both $$S$$ and $$V$$ at the same time, we also end up changing $$T$$ :

$$dT = \left(\frac{\partial T}{\partial S}\right)_V dS + \left(\frac{\partial T}{\partial V}\right)_SdV$$

and so we define $$\frac{\partial U}{\partial T}$$ to be the ratio of these two changes:

$$\frac{\partial U}{\partial T} = \frac{\left(\frac{\partial U}{\partial S}\right)_V dS + \left(\frac{\partial U}{\partial V}\right)_S dV}{ \left(\frac{\partial T}{\partial S}\right)_V dS + \left(\frac{\partial T}{\partial V}\right)_SdV}$$

Now, this quantity is not well-defined because we need to specify precisely how we change $$S$$ and $$V$$. For example, we could ask about $$\left(\frac{\partial U}{\partial T}\right)_V$$ - the rate of change of the internal energy with respect to $$T$$ when we hold $$V$$ constant. In this case $$dV=0$$ and we would find

$$\left(\frac{\partial U}{\partial T}\right)_V = \left(\frac{\partial U}{\partial S}\right)_V \big/\left(\frac{\partial T}{\partial S}\right)_V = T \left(\frac{\partial S}{\partial T}\right)_V \equiv c_V$$

where $$\left(\frac{\partial S}{\partial T}\right)_V \equiv 1\big/\left(\frac{\partial T}{\partial S}\right)_V$$ and we note the right hand side as being simply the definition of the specific heat at constant volume $$c_V$$.

If you choose some other way to change $$S$$ and $$V$$, the result will be different. For example, we could choose not to keep $$V$$ constant, but rather the pressure $$P \equiv -\left(\frac{\partial U}{\partial V}\right)_S$$.

• Can you explain more about $𝑑𝑈=𝑈(𝑆+𝑑𝑆,𝑉+𝑑𝑉)=(∂𝑈∂𝑆)𝑉𝑑𝑆+(∂𝑈∂𝑉)𝑆𝑑𝑉$ or explain what the technique is called? I looked in my calculus book and didn't find it... It looks like a gradient dotted with a vector ($\Del U \circ dL$), where $dL = dx , dy , dz$. It works with potential energy $U(x,y,z ,t)$. However, it does not work for free energy (F= U -TS) – Edward May 6 at 20:11
• @Edward If $U$ is a function of $S$ and $V$, then a small change $dS$ and a small change $dV$ will cause a small change $dU = \left(\frac{\partial U}{\partial S}\right)_VdS + \left(\frac{\partial U}{\partial V}\right)_S dV$. This is just linearization (i.e. the total differential ) applied to a function of several variables. – J. Murray May 6 at 20:15
• Thank you! The second you said that a light went off! – Edward May 6 at 20:21

If U=U(S,V), then, mathematically, you should be writing $$dU=\left(\frac{\partial U}{\partial S}\right)_VdS+\left(\frac{\partial U}{\partial V}\right)_SdV=TdS-PdV$$Treating S as a function of V and T, you also have $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV=\frac{C_v}{T}dT+\left(\frac{\partial S}{\partial V}\right)_TdV$$So, combining these two equations, you have $$dU=C_vdT-\left[P-T\left(\frac{\partial S}{\partial V}\right)_T\right]dV$$

Your first equation represents the differential of U with respect to T, as shown here: https://en.wikipedia.org/wiki/Differential_of_a_function.

This function represents a linearization about a specific point in the multi-variable equation, and I seriously doubt that it is correct to multiply that equation by dT in order to eliminate it.