For one mole the ideal gas law is pV=RT , write down the exact differential dV.
Would I be right to differentiate with respect to T and p separately?...
V = RT/p
dV = R/p - (RT)/(p^2)
I think they mean
dV = (R/p) dT - (RT)/(p^2) dp
This satisfies the requirement that the partial derivative of (R/p) with respect to p equals the partial derivative of (RT)/(p^2) with respect to T.
See
http://mathworld.wolfram.com/ExactDifferential.html
for more about exact differentials
To find the exact differential dV of the ideal gas law pV = RT, you can use partial derivatives. The exact differential of a function with multiple variables can be calculated by taking the partial derivative of the function with respect to each variable and multiplying it by the corresponding differential.
In this case, the ideal gas law has three variables: pressure (p), volume (V), and temperature (T). To find dV, we will consider it as a function of p and T while keeping V constant.
First, rearrange the ideal gas law to solve for V:
V = RT/p
Now, we will differentiate this equation with respect to both p and T separately. This means we will take the partial derivative of V with respect to p and T while treating the other variable as a constant.
∂V/∂p = -RT/p^2
∂V/∂T = R/p
Since we are treating V as a constant, the differential dV can be written as:
dV = (∂V/∂p) dp + (∂V/∂T) dT
Substituting the values we obtained from the partial derivatives:
dV = (-RT/p^2) dp + (R/p) dT
So, the exact differential dV for the ideal gas law pV = RT is given by:
dV = (-RT/p^2) dp + (R/p) dT