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A particular car does work at the rate of about\({\bf{7}}{\bf{.0}}\;{\bf{kJ/s}}\)when traveling at a steady\({\bf{21}}{\bf{.8}}\;{\bf{m/s}}\)along a level road. This is the work done against friction. The car can travel 17 km on 1.0 L of gasoline at this speed (about 40 mi/gal). What is the minimum value for\({{\bf{T}}_{\bf{H}}}\)if\({{\bf{T}}_{\bf{L}}}\)is 25°C? The energy available from 1.0 L of gas is\({\bf{3}}{\bf{.2 \times 1}}{{\bf{0}}{\bf{7}}}\;{\bf{J}}\).

Short Answer

Expert verified

The minimum value of temperature \({T_{\rm{H}}}\) is \(86\circ {\rm{C}}\).

Step by step solution

01

Understanding the efficiency of the engine

The ratio of the work done by the engine to the heat input at high temperature is termed the efficiency of the engine.

The expression for the efficiency is given as:

\(e = \frac{W}{{{Q_{\rm{H}}}}} = 1 - \frac{{{T_{\rm{L}}}}}{{{T_{\rm{H}}}}}\) … (i)

Here, W is the work done and \({Q_{\rm{H}}}\) is the heat input at high temperature.

02

Given data

The rate of doing work is \(\frac{W}{t} = 7\;{\rm{kJ/s}}\).

The speed of the car is \(v = 21.8\;{\rm{m/s}}\).

The car can travel a distance of about \(d = 17\,{\rm{km}}\).

The energy available is \(E = 3.2 \times {107}\,{\rm{J/L}}\).

The lower temperature is \({T_{\rm{L}}} = 25\circ {\rm{C}}\).

03

Evaluation of heat input from gasoline in the engine

The relation for the rate of heat input is given by:

\(\frac{Q}{t} = E \times v\)

Substitute the values in the above expression.

\(\begin{array}{l}\frac{Q}{t} = \left( {3.2 \times {{10}7}\,{\rm{J/L}} \times \frac{{1\;{\rm{L}}}}{{17000\;{\rm{m}}}}} \right) \times \left( {21.8\;{\rm{m/s}}} \right)\\\frac{Q}{t} = 41035.2\;{\rm{W}}\end{array}\)

04

Evaluation of the temperature of the engine

The relation for the efficiency is given by:

\(\begin{array}{c}\eta = \frac{{\left( {\frac{W}{t}} \right)}}{{\left( {\frac{Q}{t}} \right)}}\\1 - \frac{{{T_{\rm{L}}}}}{{{T_{\rm{H}}}}} = \frac{{\left( {\frac{W}{t}} \right)}}{{\left( {\frac{Q}{t}} \right)}}\end{array}\)

Substitute the values in the above expression.

\(\begin{array}{c}\left( {1 - \frac{{\left( {25\circ {\rm{C}} + 273} \right)\;{\rm{K}}}}{{{T_{\rm{H}}}}}} \right) = \frac{{\left( {7 \times {{10}3}\;{\rm{J/s}}} \right)}}{{\left( {41035.2\;{\rm{W}}} \right)}}\\{T_{\rm{H}}} = 359\;{\rm{K}}\\{T_{\rm{H}}} = \left( {359 - 273} \right)\circ C\\{T_{\rm{H}}} = 86\circ {\rm{C}}\end{array}\)

Thus, the minimum value of temperature\({T_{\rm{H}}}\)is\(86\circ {\rm{C}}\).

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Most popular questions from this chapter

Question:You are asked to test a machine that the inventor calls an “in-room air conditioner”: a big box, standing in the middle of the room, with a cable that plugs into a power outlet. When the machine is switched on, you feel a stream of cold air coming out of it. How do you know that this machine cannot cool the room?

(I) One liter of air is cooled at constant pressure until its volume is halved, and then it is allowed to expand isothermally back to its original volume. Draw the process on a PV diagram.

Question: (II) A heat pump is used to keep a house warm at 22°C. How much work is required of the pump to deliver 3100 J of heat into the house if the outdoor temperature is (a) 0°C, (b) \({\bf{ - 15^\circ C}}\)? Assume a COP of 3.0. (c) Redo for both temperatures, assuming an ideal (Carnot) coefficient of performance \({\bf{COP = }}{{\bf{T}}_{\bf{L}}}{\bf{/}}\left( {{{\bf{T}}_{\bf{H}}}{\bf{ - }}{{\bf{T}}_{\bf{L}}}} \right)\).

Question: (a) At a steam power plant, steam engines work in pairs, the heat output of the first one being the approximate heat input of the second. The operating temperatures of the first are 750°C and 440°C, and of the second 415°C and 270°C. If the heat of combustion of coal is \({\bf{2}}{\bf{.8 \times 1}}{{\bf{0}}^{\bf{7}}}\;{{\bf{J}} \mathord{\left/{\vphantom {{\bf{J}} {{\bf{kg}}}}} \right.} {{\bf{kg}}}}\) at what rate must coal be burned if the plant is to put out 950 MW of power? Assume the efficiency of the engines is 65% of the ideal (Carnot) efficiency. (b) Water is used to cool the power plant. If the water temperature is allowed to increase by no more than 4.5 C°, estimate how much water must pass through the plant per hour.

Question: (III) The PV diagram in Fig. 15–23 shows two possible states of a system containing 1.75 moles of a monatomic ideal gas. \(\left( {{P_1} = {P_2} = {\bf{425}}\;{{\bf{N}} \mathord{\left/{\vphantom {{\bf{N}} {{{\bf{m}}^{\bf{2}}}}}} \right.} {{{\bf{m}}^{\bf{2}}}}},\;{V_1} = {\bf{2}}{\bf{.00}}\;{{\bf{m}}^{\bf{3}}},\;{V_2} = {\bf{8}}{\bf{.00}}\;{{\bf{m}}^{\bf{3}}}.} \right)\) (a) Draw the process which depicts an isobaric expansion from state 1 to state 2, and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. (c) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume \({V_2}\), followed by an isovolumetric increase in temperature to state 2, and label this process B. (d) Find the change in internal energy of the gas for the two-step process B.

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