## Thursday, January 12, 2017

### Thermodynamics Problems with Solutions Two

We are solving series of problems on the topic thermodynamics. We need to be familiar with the concepts like specific heat and latent heat. Specific heat is the amount of heat energy required to raise the temperature of unit mass of substance by one degree centigrade or kelvin. In the case of gases, we get two different kinds of specific heats one at constant pressure and the other at constant volume. Latent heat is the amount of heat energy required to convert unit mass of substance from one state to other and it happens at constant temperature. As the heat energy is also conserved, the heat lost by a hot body is the heat gained by the cold body when the bodies are connected and there is no loss of heat energy in any format.

Problem

The ratio of specific heats of has is given to us in the problem. We need to measure the change in the internal energy of the system when volume gets doubled at constant pressure. Problem is as shown in the diagram below.

Solution

We can write the value of internal energy in terms of first law of thermodynamics and work done can be written as  the product of pressure and change in volume of the system. It can be further simplified as shown in the diagram below.

Problem

One mole of mono atomic gas is mixed with one mole of diatomic gas are mixed. We need to find the effective specific heat of the system and the problem is as shown in the diagram below.

Solution

We can write the equation for the effective specific heat by taking the conservation of energy into consideration as shown in the diagram below. We need to apply the value of specific heat of mono and diatomic gas values as per the standards and solve the problem as shown below.

Problem

Certain amount of heat is required to raise the temperature to a certain value at constant volume and we need to find out the heat energy required to raise the temperature of the system at constant pressure and the problem is as shown in the diagram below.

Solution

When the volume is constant, to find the heat energy required to raise the temperature to a certain value, we need to use the definition of specific heat of gas at constant volume. To find the heat required to raise the temperature of the system at constant pressure, we need to use the definition of specific heat of gas at constant pressure. The detailed solution is as given in the diagram below.

Problem

One mole of ideal gas expands to double to its volume under isothermal conditions at a given constant temperature as shown in the diagram below. We need to measure the work done in this case.

Solution

We know that isothermal process means the work is done at constant temperature. Here pressure is not constant and to measure the work done, we shall integrate the pressure impact under the volume conditions and we can derive a mathematical equation as we have learned. Taking that into consideration, we can solve the problem as shown in the diagram below.

Problem

For an adiabatic expansion, if a monoatomic gas expands by a certain percentage, we need to measure the percentage variation in its pressure and the problem is as shown in the diagram below.

Solution

We know that in adiabatic process, the heat energy of the system is constant and it is not going to change. Here temperature is not constant and hence Boyle’s law is not valid here. The relation between pressure and volume are different and taking that into consideration, we can solve the problem as shown in the diagram below.