An ideal gas is taken through a Carnot cycle. The isothermal...

An ideal gas is taken through a Carnot cycle. The isothermal expansion occurs at $250^{\circ} \mathrm{C},$ and the isothermal compression takes place at $50.0^{\circ} \mathrm{C}$. The gas takes in $1.20 \times 10^{3} \mathrm{J}$ of energy from the hot reservoir during the isothermal expansion. Find (a) the energy expelled to the cold reservoir in each cycle and (b) the net work done by the gas in each cycle.

Key Concepts

Carnot Cycle

The Carnot cycle is an idealized thermodynamic cycle that represents the most efficient possible heat engine operating between two temperature reservoirs. It consists of two isothermal processes where heat is absorbed and rejected, and two adiabatic processes where the system is insulated and changes temperature without heat transfer. This cycle provides a benchmark for the maximum efficiency attainable by any heat engine operating between given temperature limits.

Thermal Efficiency

Thermal efficiency is a measure of a heat engine’s ability to convert heat from a high-temperature reservoir into work. For a Carnot engine, efficiency is defined as 1 minus the ratio of the absolute temperature of the cold reservoir to that of the hot reservoir (? = 1 - Tc/Th). This concept highlights the fundamental limitation imposed by the second law of thermodynamics on the performance of all heat engines.

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout the process. In the context of the Carnot cycle, isothermal expansion occurs when the system absorbs heat at a constant high temperature, and isothermal compression occurs when heat is expelled at a constant low temperature. These processes involve heat transfer while maintaining constant internal energy for an ideal gas.

Energy Balance in Thermodynamic Cycles

The energy balance in a thermodynamic cycle is based on the first law of thermodynamics, which states that the net heat absorbed by the system over one complete cycle is equal to the net work done by the system. In a reversible cycle like the Carnot cycle, the work output is determined by the difference between the energy absorbed from the hot reservoir and the energy expelled to the cold reservoir. This concept is critical for calculating the net work and understanding energy conservation in cyclic processes.

Transcript

00:01 Since this is the carnot cycle, we can treat the efficiency and calculate it by just subtracting the hot reservoir and colder of our temperatures and dividing it by the temperature of the hot reservoir.

00:12 But we also know the efficiency is going to be equal to the work divided by the heat taken in.

00:18 And so if we calculate our efficiency through this formula, then we can solve for whatever we need here.

00:24 Now we also know work is going to be equal to qh minus qc, right, the change in the energy.

00:30 So we can rewrite it like this.

00:32 And then we can solve for whatever we need.

00:36 The first part of the question is asking for qc, and the second is for work, but we can just do it more simply if we just find work first and then find qc using, you know, just defining the difference.

00:47 So let's find efficiency first.

00:49 That's the first step.

00:50 So what's it going to be? th minus tc over th.

00:52 Now, we are given the temperatures, but they are not in kelvin.

00:58 So we have th is equal to 250 degrees celsius.

01:01 We add 273 to that to get our temperature in kelvin, which is 523.

01:08 Tc, we're told, is 50 degrees celsius.

01:12 We add 273 to that to get our temperature in kelvin, and that's going to be 323 kelvin.

01:19 So now let's plug those numbers in...