Thermodynamics: Show that the two relations give Pv = RT

[karan4496]

Homework Statement

For an ideal gas the slope of an isotherm is given by

(∂P/∂v) constant T = -P/v

and that of an isochore is

(∂P/∂T) constant v = P/T

Show that these relations give Pv = RT

Homework Equations

Pv = RT

The Attempt at a Solution

I have never worked with partial derivatives before encountering this problem so I am unfamiliar with the rules and operations involved. I tried setting them equal, adding them to each other but I just don't know where I am going.

 

[voko]

You can treat each of the equations as an ordinary differential equation where the independent variable is V and T respectively. When you solve them, you will have two constants. But these constants must be then functions of the "constant" variable, T and V respectively. Then you should be able to find those functions and get the ideal gas law.

 

[karan4496]

Ok so I got up to this step.

dP = -P/v dv + P/T dt

Im unsure of how to proceed from here

 

[vela]

Divide by P and then integrate.

 

[paranormal]

Dear student,

The two relations given (∂P/∂v) constant T = -P/v and (∂P/∂T) constant v = P/T are both partial derivatives of the ideal gas law, which is given by Pv = RT. This law describes the relationship between pressure (P), volume (v), temperature (T), and the universal gas constant (R) for an ideal gas.

To show that these two relations give Pv = RT, we can use the chain rule for partial derivatives. Starting with the first relation, we can write:

(∂P/∂v) constant T = (∂P/∂v) constant T * (∂v/∂v) constant T

= (∂P/∂v) constant T * 1 (since volume is constant)

= (∂P/∂v) constant T

Similarly, for the second relation, we can write:

(∂P/∂T) constant v = (∂P/∂T) constant v * (∂T/∂T) constant v

= (∂P/∂T) constant v * 1 (since temperature is constant)

= (∂P/∂T) constant v

Now, we can substitute these expressions into the ideal gas law, Pv = RT, to get:

(∂P/∂v) constant T * (∂v/∂v) constant T * v = (∂P/∂T) constant v * (∂T/∂T) constant v * v

Simplifying, we get:

(∂P/∂v) constant T * v = (∂P/∂T) constant v * v

Canceling out the v on both sides, we are left with:

(∂P/∂v) constant T = (∂P/∂T) constant v

Substituting back into the original equations, we get:

- P/v = P/T

Multiplying both sides by v and T, we get:

- Pv = PT

But we know that from the ideal gas law, Pv = RT, so we can substitute this in to get:

- RT = PT

Dividing both sides by T, we get:

- R = P

Since the gas constant (R) and pressure (P) are both constants, we can simplify this to