Question

Question #d1389

Answer 1

For ideal gas on a pV diagram, an adiabatic curve is always steeper than an isotherm if they intersect.

Explanation:

The ideal gas equation: `pV = nRT`

• `p` is pressure
• `V` is volume
• `n` is number of moles
• `R` is ideal gas constant
• `T` is the absolute temperature

For an isothermal process, the right hand side of the ideal gas equation does not change. Which means

`pV = "constant"`

The slope at any point is given by

`{dp}/{dV} = -p/V`

using implicit differentiation or any other methods.

For adiabatic processes, the path would follow a curve of the following form:

`pV^gamma = "constant"`

`gamma` in the above equation is treated as a constant and is meant to be the ratio of heat capacities at constant pressure, `c_p`, to heat capacity at constant volume, `c_V`. This can be derived using the first Law of Thermodynamics.

Differetiating to find the slope

`{dp}/{dV} = -gamma p / V`

This looks very similar to the expression for the slope of isotherm, except that gamma is always > 1 and therefore the slope would be steeper.