Problem 38 - University Physics with Modern Physics

Question

20.38. You are designing a Carnot engine that has 2 \(\mathrm{mol}\) of \(\mathrm{CO}_{2}\) as its working substance; the gas may be treated as ideal. The gas is to have a maximum temperature of \(527^{\circ} \mathrm{C}\) and a maximum pressure of 5.00 \(\mathrm{atm}\) . With a heat input of 400 \(\mathrm{J}\) per cycle, you want 300 \(\mathrm{J}\) of useful work (a) Find the temperature of the cold reservoir. (b) For how many cycles must this engine run to melt completely a 10.0 kg block of ice originally at \(0.0^{\circ} \mathrm{C}\) , using only the beat rejected by the engine?

Step-by-Step Solution

Answer

(a) 200.0375 K (b) 33,400 cycles

Step 1: Analyze the Carnot Engine Efficiency

The efficiency of a Carnot engine is given by \( \eta = 1 - \frac{T_c}{T_h} \), where \( T_c \) is the temperature of the cold reservoir and \( T_h \) is the temperature of the hot reservoir. We convert the maximum temperature from Celsius to Kelvin: \( T_h = 527 + 273.15 = 800.15 \text{ K} \). The efficiency can also be expressed in terms of work and heat input: \( \eta = \frac{W}{Q_h} \). Given \( W = 300 \text{ J} \) and \( Q_h = 400 \text{ J} \), we find \( \eta = \frac{300}{400} = 0.75 \).

Step 2: Calculate Cold Reservoir Temperature

From the efficiency formula \( \eta = 1 - \frac{T_c}{T_h} \), we can rearrange it to solve for \( T_c \): \( T_c = T_h (1 - \eta) \). Substituting in the known values: \( T_c = 800.15 (1 - 0.75) = 800.15 \times 0.25 = 200.0375 \text{ K} \).

Step 3: Determine Heat Rejected

The heat rejected \( Q_c \) can be found using the formula \( Q_c = Q_h - W \). Given \( Q_h = 400 \text{ J} \) and \( W = 300 \text{ J} \), we have \( Q_c = 400 - 300 = 100 \text{ J} \).

Step 4: Calculate Number of Cycles to Melt Ice

To melt the ice, we use the latent heat of fusion \( L_f = 334,000 \text{ J/kg} \) for ice. The heat required to melt 10.0 kg of ice is \( Q_{ice} = m L_f = 10 \times 334,000 = 3,340,000 \text{ J} \). The number of cycles needed is \( n = \frac{Q_{ice}}{Q_c} = \frac{3,340,000}{100} = 33,400 \) cycles.

Key Concepts

Ideal Gas

Ideal gases are theoretical models that help us understand the behavior of gas molecules under different conditions. An ideal gas behaves predictably using certain assumptions, particularly at low pressures and high temperatures.

Key characteristics of an ideal gas include:

• Molecules move randomly and continuously.
• Molecules are point particles, meaning they have negligible volume.
• No intermolecular forces act between them except during collisions.
• Collisions are perfectly elastic, conserving both momentum and energy.

The ideal gas law, expressed as \( PV = nRT \), relates pressure \( P \), volume \( V \), number of moles \( n \), the universal gas constant \( R \), and temperature \( T \). This allows us to predict how a gas will respond to changes in these variables under the assumption of ideal behavior.

Carnot Cycle

The Carnot cycle is a theoretical model that represents the most efficient way to convert heat into work and vice versa. Formulated by Sadi Carnot, it is an iterative process involving four reversible steps: two isothermal processes and two adiabatic processes.

The cycle operates as follows:

• Isothermal Expansion: The gas absorbs heat from a hot reservoir at a constant high temperature \( T_h \), expanding and doing work.
• Adiabatic Expansion: The gas expands further without heat exchange, lowering its temperature to that of the cold reservoir \( T_c \).
• Isothermal Compression: The gas expels heat to the cold reservoir at a constant temperature \( T_c \), doing work on the gas.
• Adiabatic Compression: The gas is compressed further without heat exchange, raising its temperature back to \( T_h \).

Carnot's theorem states that no real engine operating between two heat reservoirs can be more efficient than a Carnot engine.

Heat Engine Efficiency

Heat engine efficiency refers to the ratio of the work output of a heat engine to the heat input. It is a measure of how well an engine converts heat into useful work. For real-world engines, many factors such as friction, heat losses, and imperfect processes lead to less than maximum efficiency. However, the Carnot engine sets an idealized benchmark for efficiency.

The formula for heat engine efficiency is:\[\eta = \frac{W}{Q_h} = 1 - \frac{T_c}{T_h}\]Where:

• \( W \) is the work done by the engine.
• \( Q_h \) is the heat absorbed from the hot reservoir.
• \( T_c \) and \( T_h \) are the temperatures of the cold and hot reservoirs, respectively.

Perfect efficiency is impossible due to the second law of thermodynamics, which implies that some heat will always be expelled during the process. The Carnot cycle serves as the upper limit to which real engines can aspire.