Given, an ideal gas of bulk modulus 'K' and is heated to a temperature .
Increase of temperature of gas at constant volume is equivalent if the gas is first heated at constant pressure and then compressed isothermally to original volume.
Let V be the volume of gas. When it is heated through a temperature θ at constant pressure, then the increase in volume of gas is,
∆V = Vγθ
Now compress the gas isothermally so that the volume decreases by ∆V = Vγθ.
If ∆P be the stress set up in the gas, then
When a rod of metal is heated, compressive stress will be set up in the rod. The wire is not allowed to bend and the force is applied normal to it's cross sectional area.
Let 'L' be the length of wire, A be the area of cross section, Y be Young's modulus and '' be the coefficient of linear expansion, heated to a temperature ''.
If the rod was free then increase in length of wire due to rise in the temperature will be Lθ.
Here, since rod is not allowed to expand, therefore, compression in rod is equal to increase in the length of rod i.e. Lθ.
If S is the compression stress in rod then,
Hence stress is independent of the length of the wire.