A uniform pressure P is exerted on all sides of a solid cube at t0C. By what amount should the temperature of the cube be raised in order to bring its volume back to the value it had before the pressure was applied ? ( K is bulk modulus and α is coefficient of linear expansion)
Question
A uniform pressure P is exerted on all sides of a solid cube at t0C. By what amount should the temperature of the cube be raised in order to bring its volume back to the value it had before the pressure was applied ? ( K is bulk modulus and α is coefficient of linear expansion)
Solution
The volume of a solid changes with pressure and temperature. The change in volume due to pressure can be given by the formula ΔV = -PΔV/K, where K is the bulk modulus. The change in volume due to temperature can be given by the formula ΔV = 3αVΔT, where α is the coefficient of linear expansion and ΔT is the change in temperature.
In this case, the pressure is causing a decrease in volume, and we want to find the increase in temperature that will cause an equivalent increase in volume, so we set the two equations equal to each other and solve for ΔT:
-PΔV/K = 3αVΔT
Solving for ΔT gives:
ΔT = -PΔV/(3αVK)
Since we want the volume to return to its original value, ΔV = 0, so the equation simplifies to:
ΔT = -P/(3αK)
This is the amount the temperature needs to be increased to bring the volume of the cube back to its original value.
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