Work Done In Thermodynamics

Given,

Number of moles

Initial volume



Final volume



Temperature

Work done due to change in volume against a constant pressure is given by





or

The work done by a gas during an isobaric process (constant pressure) is given by the formula: Where: - is the work done by the gas, - is the pressure, - is the change in volume. Thus, the work done is: The negative sign indicates that the gas is being compressed, meaning the work done on the gas is negative. Therefore, the work done by the gas is .

Step 1: Understanding the Boltzmann distribution. In thermodynamics, the probability that a system's particle has energy is given by the Boltzmann distribution: where: - is the energy of the particle, - is the Boltzmann constant, - is the temperature of the system, - is the partition function, which normalizes the probability distribution. The Boltzmann distribution shows how the probability of a particle having a certain energy decreases exponentially with increasing energy. Step 2: Finding the probability of energy greater than . We are interested in the probability that the energy of a particle is greater than . This probability is the complement of the probability that the particle has energy less than or equal to : From the Boltzmann distribution, the probability that the particle has energy less than or equal to is: The result of this integral gives the probability that the energy is less than , and hence the probability that the energy is greater than is: Step 3: Conclusion. Thus, the probability that a particle will have energy greater than according to the Boltzmann distribution is: Answer: Therefore, the probability is .

For the given process , the work done by the gas during the expansion is given by the integral: Since , the work done can be calculated as: Using the limits and , and knowing the relationship , we can calculate the work done. The pressure at can be found from the initial condition: Thus, the work done by the gas is approximately 10.1 J.