Meaning of subscript in partial derivative notation

[kaashmonee]

Homework Statement

I'm given a gas equation, $PV = -RT e^{x/VRT}$, where $x$ and $R$ are constants. I'm told to find $\Big(\frac{\partial P}{\partial V}\Big)_T$. I'm not sure what that subscript $T$ means?

Homework Equations

$PV = -RT e^{x/VRT}$

Thanks a lot in advance.

 

[fresh_42]

Homework Statement

I'm given a gas equation, $PV = -RT e^{x/VRT}$, where $x$ and $R$ are constants. I'm told to find $\Big(\frac{\partial P}{\partial V}\Big)_T$. I'm not sure what that subscript $T$ means?

Homework Equations

$PV = -RT e^{x/VRT}$

Thanks a lot in advance.

Firstly, in order to make sense of the notation $\dfrac{\partial P}{\partial V}$ it seems, that $P$ is a function of $V$ and other variables. I assume they all are pressure, volume and temperature. Usually I would say $\left( \dfrac{\partial P}{\partial V} \right)_T$ would mean $\left( \left. \dfrac{\partial P}{\partial V}\right|_{V=V_0} \right)$ which is the partial derivative evaluated at the point $V=V_0$. But as temperature $T$ is hardly a volume, it means probably something else in this context. My guess would simply be $\left( \dfrac{\partial P}{\partial V} \right)_T = \dfrac{\partial}{\partial V} P(V,T)$ stretching that $T$ is constant here for the differentiation, but a variable for the resulting derivative. If evaluated at say $V=V_0$, then it is $\left( \dfrac{\partial P}{\partial V} \right)_T= \left( \left. \dfrac{\partial P}{\partial V}\right|_{V=V_0} \right) (V_0,T)$, i.e. the partial derivative at a certain point $V_0$ for the volume, leaving it a function of temperature $T$.

 

[Mark44]

I'm going to disagree with @fresh_42 here. It most likely means the partial of P with respect to V, but holding T constant.

From Wikpedia, https://en.wikipedia.org/wiki/Partial_derivative#Notation,

When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics, the partial derivative of $f$ with respect to $x$, holding $y$ and $z$ constant, is often expressed as
$$\left(\frac{\partial f}{\partial x}\right)_{y, z} $$

For the partial you're asking about, you could rewrite the equation as $P = \frac{-RTe^{x/V RT}}{V}$. I'm not sure exactly what your equation is supposed to be as it's ambiguous -- is the exponent on e $\frac x {VRT}$ or is it $\frac x V RT$? If it's the former, it should have been written as $e^{x/(VRT)}$. If it's the latter, it would be clearer as $e^{(x/V)RT}$.
In any case, you now have P as the dependent variable, and V as the only independent variable, with x, R, and T being or acting as constants.

 

[Marc Rindermann]

Homework Statement

I'm given a gas equation, $PV = -RT e^{x/VRT}$, where $x$ and $R$ are constants. I'm told to find $\Big(\frac{\partial P}{\partial V}\Big)_T$. I'm not sure what that subscript $T$ means?

Thanks a lot in advance.

it means that you evaluate the differentiation and keep $T$ constant. Similarly you could find $\left(\frac{\partial P}{\partial T}\right)_V$ where you would treat $V$ as a constant