Finding the heat transferred in an ininitesimal quasistatic process

[Narcol2000]

For an ideal gas PV=nRT where n is the number of moles show that the heat transferred can be written as:

$$dQ = \frac{C_V}{nR}VdP + \frac{C_P}{nR}PdV$$

Really not sure where to start with this...

I have used

$$dQ = dU + PdV$$

But it hasn't really lead anywhere.

 

[Andrew Mason]

For an ideal gas PV=nRT where n is the number of moles show that the heat transferred can be written as:

$$dQ = \frac{C_V}{nR}VdP + \frac{C_P}{nR}PdV$$

Really not sure where to start with this...

I have used

$$dQ = dU + PdV$$

But it hasn't really lead anywhere.

Start with:

dQ = mCdT where C is the heat capacity of the gas. (The heat capacity, of course, depends on whether the gas does work as the heat flows into the gas, so it does not have a fixed value).

Using PV = nRT, what is dT? That should get you going in the right direction.

AM

 

[astrokreng]

I can provide some insights and explanations on how to approach this problem. First, we need to understand the concept of heat transfer in a thermodynamic process. Heat transfer is the transfer of energy from one system to another due to a temperature difference between them.

In this case, we are dealing with an infinitesimal quasistatic process, which means that the process is happening very slowly and the system is always in thermodynamic equilibrium. This allows us to use the ideal gas law, PV=nRT, to describe the behavior of the gas.

Now, let's break down the equation for heat transfer in this process. We know that the change in internal energy, dU, is equal to the heat transferred, dQ, plus the work done, PdV. So, we can rewrite the equation as:

dQ = dU - PdV

Next, we can use the ideal gas law to substitute for dU and PdV:

dQ = nC_VdT - nRdT = n(C_V - R)dT

Now, we need to relate the change in temperature, dT, to the changes in pressure, dP, and volume, dV. This can be done by using the chain rule:

dT = \frac{\partial T}{\partial P}dP + \frac{\partial T}{\partial V}dV

Substituting this into our equation for dQ, we get:

dQ = n(C_V - R)(\frac{\partial T}{\partial P}dP + \frac{\partial T}{\partial V}dV)

Next, we can use the ideal gas law again to relate the changes in temperature to changes in pressure and volume:

\frac{\partial T}{\partial P} = \frac{nR}{C_PV} and \frac{\partial T}{\partial V} = -\frac{nR}{C_VV}

Substituting these into our equation, we get:

dQ = n(C_V - R)(\frac{nR}{C_PV}dP - \frac{nR}{C_VV}dV)

Simplifying, we get:

dQ = \frac{nR}{C_PV}C_VdP - \frac{nR}{C_VV}C_PdV

Finally, we can use the definition of specific heat capacity, C = \frac{dQ