| 3 |
 |
| 2 |
| 3 |
|
| 2 |
A constant volume process is also known as an isochoric process. An example is when heat is added to a gas in a container with fixed walls.
Because the walls can't move, the gas can not do work:
W = 0
In that case the First Law states:
Q = ΔEint
The P-V diagram for this process is simple - it's a vertical line going up if heat is added, and going down if heat is removed.
In the case of a monatomic ideal gas:
| Eint | = |
|
NkT | = |
|
nRT |
| Therefore Q | = | ΔEint | = |
|
nRΔT |
A constant temperature process is an isothermal process. An example is when a gas in a container that is immersed in a constant-temperature bath is allowed to expand slowly, or is compressed slowly.
At constant temperature there is no change in internal energy.
ΔEint = 0
Apply the First Law:
Q = W
The P-V diagram for this process follows an isotherm, a line of constant temperature.
For an ideal gas at constant temperature, the pressure is inversely proportional to the volume:
| P | = |
|
, so: |
| W | = | ∫ | P dV | = nRT | ∫ |
|
dV |
The integral of 1/V is ln(V), and ln(A)-ln(B) = ln(A/B).
| Therefore: Q = W | = nRT ln | ( |
|
) |
A constant pressure process is called an isobaric process. An example is a gas in a container sealed with a piston that is free to slide up and down.
If heat is added the temperature goes up and the system expands, so work is done.
The full First Law applies:
ΔEint = Q - W
The P-V diagram for this process is a horizontal line, so the work done is simply:
W = P ΔV = nR ΔT
For a monatomic ideal gas:
| ΔEint | = |
|
nR ΔT |
Plugging this into the First Law gives:
Q = ΔEint + W
| Q | = |
|
nR ΔT | + | nR ΔT | = |
|
nR ΔT |