A monatomic ideal gas is taken around the cycle shown in...

A monatomic ideal gas is taken around the cycle shown in Fig. $P 20.51$ in the direction shown in the figure. The path for process $c \rightarrow a$ is a straight line in the $p V$ -diagram. (a) Calculate $Q,$ $W,$ and $\Delta U$ for each process $a \rightarrow b, b \rightarrow c,$ and $c \rightarrow a$ . (b) What are $Q, W,$ and $\Delta U$ for one complete cycle? (c) What is the efficiency of the cycle?

Key Concepts

First Law of Thermodynamics

This principle states that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system. It is the foundation for analyzing any thermodynamic process or cycle, establishing energy conservation in transformations between different forms of energy.

Internal Energy of a Monatomic Ideal Gas

For a monatomic ideal gas, the internal energy depends solely on its temperature, and can be calculated using the formula ?U = (3/2)nR?T. This concept is crucial since it links microscopic thermal motion with macroscopic thermodynamic properties and simplifies the analysis of energy changes during various processes.

Work in p-V Diagrams

Work done by or on a system during a thermodynamic process is represented by the area under or enclosed by a path in a pressure-volume (p-V) diagram. Understanding this concept helps in calculating work for different paths, especially in cyclic processes where the net work corresponds to the cycle's area.

Heat Transfer in Thermodynamic Processes

Heat (Q) is the energy exchanged due to temperature differences during a process. Its evaluation, along with work and internal energy change, requires careful attention to the specific path of the process, as heat is a path function in contrast to state functions like internal energy.

Efficiency of Thermodynamic Cycles

Efficiency is defined as the ratio of net work output to the heat input of the cycle. It measures how effectively a thermodynamic cycle converts heat into work and is essential in assessing the performance of heat engines and other cyclic processes.

Transcript

00:01 At for the given question, so here from the diagram we have to calculate in this, we have to calculate q, w and delta u, which have the traditional meaning of q, w and delta u.

00:18 And it is given cp is equal to 5 upon 2r and specificate capacity at constant volume.

00:31 It is given cv is equal to 3 upon 2r.

00:35 Cp is specific head capacity of the gas at constant pressure.

00:41 So in the first bit we have to calculate q, w and delta u.

00:49 For the process ab, a to b.

00:55 In a b process, delta p is constant.

01:01 Constant.

01:05 So we know that q is equal to n, cp delta t.

01:11 But from the ideal gas equation, pv is equal to to n r t so from this we can we can say that n into delta t is equal to p into delta v upon r now we will substitute the value in this so q is equal to p delta v upon r into cp now we will substitute the value from the diagram so here pressure is given 3 .0 into 10 to the over 5 and volume is increasing from 0 .5 meter cube to 0 .8 meter cube into cps 5 .2 into r upon r.

02:10 So this r and r will be cancelled to each other.

02:14 So after calculation we will get 2 .25 into 10 to the power 5 jr.

02:30 Now we will calculate work during the process ab.

02:38 So w is equal to p into delta v.

02:43 So here volume is getting increased.

02:46 Now we will substitute the value from the diagram.

02:48 So 3 into 10 to the power 5 into 0 .8 final volume minus initial volume.

03:00 Now after solving we will get 0 .9 into 10 to the power 5.

03:06 Power 5 joules...