Question

Question: A refrigerator has a coefficient of performance of \(2.10\). In each cycle it absorbs \(3.10 \times 1{0^4}\;J\) of heat from the cold reservoir. (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat is discarded to the high-temperature reservoir?

Short answer

The mechanical energy is each cycle required to operate the refrigerator is\(1.47 \times {10^4}\;{\rm{J}}\).

Step-by-step solution

Write the given data from the question.

The coefficient of performance,\(K = 2.10\)

The absorbs heat, \({Q_C} = 3.10 \times {10^4}\;{\rm{J}}\)

Determine the formulas to calculate the mechanical energy is each cycle required to operate the refrigerator.

The coefficient of performance is defined as the ratio of the absorbs heat and work done.

The expression to calculate the coefficient of performance is given as follows.

\(K = \left| {\frac{{{Q_C}}}{W}} \right|\) …… (i)

Here,\({Q_C}\)is the heat absorbs and\(W\)is the work done.

Calculate the mechanical energy is each cycle required to operate the refrigerator.

Derive the expression for the work done from the equation (i).

\(W = \frac{{\left| {{Q_C}} \right|}}{K}\)

Substitute \(2.10\) for \(K\) and \(3.10 \times {10^4}\;{\rm{J}}\) for \({Q_C}\) into above equation.

\(\begin{array}{l}W = \frac{{3.10 \times {{10}^4}\;{\rm{J}}}}{{2.10}}\\W = \frac{{310 \times {{10}^4}\;{\rm{J}}}}{{210}}\\W = 1.47 \times {10^4}\;{\rm{J}}\end{array}\)

Hence the mechanical energy is each cycle required to operate the refrigerator is\(1.47 \times {10^4}\;{\rm{J}}\).