Question

An ideal Carnot engine operates between 500\(^\circ\)C and 100\(^\circ\)C with a heat input of 250 J per cycle. (a) How much heat is delivered to the cold reservoir in each cycle? (b) What minimum number of cycles is necessary for the engine to lift a 500-kg rock through a height of 100 m?

Short answer

(a) 120.75 J; (b) At least 3797 cycles.

Step-by-step solution

Convert Temperatures from Celsius to Kelvin

The temperatures given in the problem need to be converted from Celsius to Kelvin because temperature calculations must be in Kelvin. The conversion formula is \( T(K) = T(°C) + 273.15 \). So, for 500°C, \( T_1 = 500 + 273.15 = 773.15 \) K. For 100°C, \( T_2 = 100 + 273.15 = 373.15 \) K.

Calculate Efficiency of Carnot Engine

The efficiency of a Carnot engine is given by \( \eta = 1 - \frac{T_2}{T_1} \), where \( T_1 \) and \( T_2 \) are the temperatures of the hot and cold reservoir, respectively. Substituting the temperatures: \( \eta = 1 - \frac{373.15}{773.15} \approx 0.517 \).

Calculate Work Done by the Engine

The work done \( W \) by the engine can be calculated using the formula \( W = Q_1 \times \eta \), where \( Q_1 \) is the heat input. Given \( Q_1 = 250 \) J and \( \eta \approx 0.517 \), we find \( W = 250 \times 0.517 \approx 129.25 \) J.

Calculate Heat Delivered to Cold Reservoir

The heat delivered to the cold reservoir \( Q_2 \) can be found using the relation \( Q_1 = W + Q_2 \). Rearranging gives \( Q_2 = Q_1 - W \). Therefore, \( Q_2 = 250 - 129.25 \approx 120.75 \) J.

Calculate Work Required to Lift a Rock

The work required to lift a rock is calculated using the formula \( W = mgh \), where \( m \) is the mass of the rock, \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)) and \( h \) is the height. Here, \( m = 500 \) kg and \( h = 100 \) m, so \( W = 500 \times 9.81 \times 100 = 490500 \) J.

Calculate Number of Cycles Required

To find the number of cycles needed to do 490500 J of work, divide the total work by the work per cycle. Using the previously calculated work per cycle (129.25 J), the number of cycles \( n \) is \( n = \frac{490500}{129.25} \approx 3796.5 \). Thus, at least 3797 cycles are required, since we count partial cycles as full.

Key concepts

Thermodynamic Efficiency

The concept of thermodynamic efficiency relates to how well a Carnot engine converts heat into work. For an ideal Carnot engine, this efficiency is given by the equation \( \eta = 1 - \frac{T_2}{T_1} \), where \( T_1 \) and \( T_2 \) are the absolute temperatures of the hot and cold reservoirs, respectively.
This efficiency equation highlights a key principle: the efficiency of a Carnot engine depends directly on the temperature difference between the two reservoirs.
The larger this temperature difference, the higher the potential efficiency. However, it's worth noting that even with perfect, theoretical conditions, no engine can achieve 100% efficiency due to the second law of thermodynamics. The Carnot cycle represents the maximum possible efficiency any engine operating between two temperatures can achieve.

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics, playing a crucial role in the operation of heat engines like the Carnot engine. In the context of the exercise, heat transfer occurs primarily between the heat source (hot reservoir) and the cold reservoir.
In each cycle of the Carnot engine described, a certain amount of heat, \( Q_1 = 250 \) J, is absorbed from the hot reservoir. This heat serves as the engine's input.
The engine converts part of this heat into work, with the rest being transferred to the cold reservoir. The heat delivered to the cold reservoir can be calculated using \( Q_2 = Q_1 - W \), where \( W \) is the work done.

• A well-designed engine maximizes \( W \), trying to keep \( Q_2 \) as low as practically possible.
• Effective heat transfer is essential for maintaining efficiency.

This balance between input heat, work conversion, and residual heat transfer illustrates the constraints an ideal engine faces.

Work and Energy

In the context of a Carnot engine, work and energy are interconnected concepts that describe the engine's purpose and operation. The energy provided as heat is transformed into work across each cycle of the engine.
The engine's job is to perform useful work, such as lifting a weight, as described in the exercise. This is calculated by the formula \( W = mgh \), where \( m \) is the mass of the object, \( g \) is gravitational acceleration, and \( h \) is the height lifted.
In the exercise, the engine must perform a substantial amount of work, 490,500 J, to lift a rock. To accomplish this, the Carnot engine must undergo several cycles, each contributing a portion of the necessary energy.

• The total energy required depends on the mass of the object and the height it is lifted.
• The number of cycles required is determined by dividing the total energy by the work output per cycle.

In an ideal situation, balancing the conversion of heat into work dictates how efficiently energy is used in performing tasks.