Question

An ideal gas heat engine is operating between \(227^{\circ} \mathrm{C}\) and \(127^{\circ} \mathrm{C}\). It absorbs \(10^{4} \mathrm{~J}\) Of heat at the higher temperature. The amount of heat Converted into. work is \(\ldots \ldots\) J. (A)2000 (B) 4000 (C) 5600 (D) 8000

Short answer

The amount of heat converted into work by the ideal gas heat engine is \(\textbf{2000 J}\) (A).

Step-by-step solution

Convert Temperatures to Kelvin

First, we need to convert the given temperatures from Celsius to Kelvin. To do this, add 273.15 to both temperatures. \(T_H = 227 + 273.15 = 500.15 K\) \(T_L = 127 + 273.15 = 400.15 K\)

Calculate Efficiency of the Heat Engine

Next, we can use the Carnot efficiency formula for heat engines: \(Efficiency = 1 - \frac{T_L}{T_H}\) Plug in the values for \(T_H\) and \(T_L\). \(Efficiency = 1 - \frac{400.15}{500.15}\) \(Efficiency = 1 - 0.80 = 0.20\)

Calculate the Amount of Heat Converted into Work

We can now use the definition of efficiency in terms of heat absorbed and heat converted into work: \(Efficiency = \frac{Work}{Heat\, Absorbed}\) Rearrange this equation to solve for the amount of work done: \(Work = Efficiency \times Heat\, Absorbed\) Now, plug in the values for efficiency and heat absorbed: \(Work = 0.20 \times 10^4 J\) \(Work = 2,000 J\)

Match the Answer to Options

The amount of heat converted into work is 2,000 J, which matches option (A). So, the final answer is: (A) 2000 J