Activity: First Law and thermodynamic identity

Prerequisite concepts

Differentials are small changes or differences

One typical notation used for derivatives is Leibniz notation ($e.g.,$ $df/dx$). Physicists often think of the $df$ and $dx$ as distinct quantities (``differentials'') that can be measured, calculated, and manipulated independent of one another. In particular, a differential can be thought of as the (small) change in a quantity when a small change is made to a physical system. A differential can also be thought of as the difference between the values of a quantity between two different (nearby) physical states. This line of thinking is also valuable for thinking about integration in physical situations.

Equations can be 'zapped with d' to relate differentials Individual differentials can be manipulated algebraically Total differentials are linear

Energy and Integrals

Differentials are small chunks

$df$ can be viewed as a small chunk of $f$. When combined with 'The derivative is a ratio of small changes', it's easy to relate differentials to derivatives.

There might be experimental limits on which quantities you can measure

Partial Derivative Machine Derivatives

The coefficients in a differentials equation are partial derivatives

FIXME

There are experimental limits to how $small$ of a change can be measured The derivative can be interpreted physically Partial derivatives that do not have the same variable(s) held constant (at the same values?) are not the same derivative

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First Law and thermodynamic identity

The First Law[\Delta U=Q+W][dU=dQ+dW]TheThermodynamic Identity[]The internal energy is clearly a state function, and thus its differential must be an exact differential [𝑑𝑈= ?=đ𝑄−đ𝑊=đ𝑄−𝑝𝑑𝑉] only when change is quasistatic. This $−𝑝𝑑𝑉$ term can be a bit confusing at first. You are accustomed to work being $𝐹𝑑𝑥$. With a little thought, you can recognize pp as the force per unit volume, and the ratio of 𝑑𝑉 and 𝑑𝑥 as the area. The minus sign comes from the fact that a positive pressure pushes outwards. What is this $đ𝑄$? As it turns out, we can define a state function 𝑆 called entropy and so long as a process is done reversibly $đ𝑄=𝑇𝑑𝑆$ only when change is quasistaticso we find out that𝑑𝑈=𝑇𝑑𝑆−𝑝𝑑𝑉The fact that the 𝑇 in this equation is actually the physical temperature measured by our thermometers was originally an observation based on experiment. At this point, the entropy 𝑆 is just some weird heat-related state function.If you decide to get a thermodynamics tattoo, my recommendation would be to choose the thermodynamic identity [𝑑𝑈=𝑇𝑑𝑆−𝑝𝑑𝑉] It is far and away the most fundamental and essential equation, and one which you will need to come back to again and again. It contains hidden within it (if you remember the First Law) the thermodynamic definition of entropy.

Representations used

$df$

$\left(\frac{\partial f}{\partial x}\right)_y$

Concepts taught

The thermodynamic identity defines temperature and pressure as partial derivatives

[T = \left(\frac{\partial U}{\partial S}\right)_V] [p = -\left(\frac{\partial U}{\partial V}\right)_S]

$đQ = TdS$ (edit later)

ENTER LONG DESCRIPTION

$đW = - pdV$ (edit later)

ENTER LONG DESCRIPTION