Problems & Exercises

What is the change in internal energy of a car if you put 12.0 gal of gasoline into its tank? The energy content of gasoline is $1\text{.}3\times {\text{10}}^8\text{J/gal}$. All other factors, such as the car’s temperature, are constant.

A system does $1\text{.}\text{80}\times {\text{10}}^8\text{J}$ of work while $7\text{.}\text{50}\times {\text{10}}^8\text{J}$ of heat transfer occurs to the environment. What is the change in internal energy of the system assuming no other changes (such as in temperature or by the addition of fuel)?

Suppose a woman does 500 J of work and 9500 J of heat transfer occurs into the environment in the process. (a) What is the decrease in her internal energy, assuming no change in temperature or consumption of food? (That is, there is no other energy transfer.) (b) What is her efficiency?

(a) What is the average metabolic rate in watts of a man who metabolizes 10,500 kJ of food energy in one day? (b) What is the maximum amount of work in joules he can do without breaking down fat, assuming a maximum efficiency of 20.0%? (c) Compare his work output with the daily output of a 187-W (0.250-horsepower) motor.

(a) A woman climbing the Washington Monument metabolizes $6\text{.}\text{00}\times {\text{10}}^2\text{kJ}$ of food energy. If her efficiency is 18.0%, how much heat transfer occurs to the environment to keep her temperature constant? (b) Discuss the amount of heat transfer found in (a). Is it consistent with the fact that you quickly warm up when exercising?

A car tire contains $0\text{.}\text{0380}{m}^3$ of nitrogen at a gauge pressure of $2\text{.}\text{20}\times {\text{10}}^5{\text{N/m}}^2$ (about 32 psi). How much more internal energy does this gas have than the same volume has at zero gauge pressure (which is equivalent to normal atmospheric pressure)? (Use $E_{\text{int}}=\frac{5}{2}NkT$ for the internal energy of nitrogen, a diatomic gas.)

Steam to drive an old-fashioned steam locomotive is supplied at a constant gauge pressure of $1\text{.}\text{75}\times {\text{10}}^6{\text{N/m}}^2$ (about 250 psi) to a piston with a 0.200-m radius. (a) By calculating $P\text{Δ}V$, find the work done by the steam when the piston moves 0.800 m. Note that this is the net work output, since gauge pressure is used. (b) Now find the amount of work by calculating the force exerted times the distance traveled. Is the answer the same as in part (a)?

Calculate the net work output of a heat engine following path ABCDA in the figure below.

Figure 15.41

Unreasonable Results

What is wrong with the claim that a cyclical heat engine does 4.00 kJ of work on an input of 24.0 kJ of heat transfer while 16.0 kJ of heat transfers to the environment?

A certain heat engine does 10.0 kJ of work and 8.50 kJ of heat transfer occurs to the environment in a cyclical process. (a) What was the heat transfer into this engine? (b) What was the engine’s efficiency?

(a) What is the work output of a cyclical heat engine having a 22.0% efficiency and $6\text{.}\text{00}\times {\text{10}}^9\text{J}$ of heat transfer into the engine? (b) How much heat transfer occurs to the environment?

The engine of a large ship does $2\text{.}\text{00}\times {\text{10}}^8\text{J}$ of work with an efficiency of 5.00%. (a) How much heat transfer occurs to the environment? (b) How many barrels of fuel are consumed, if each barrel produces $6\text{.}\text{00}\times {\text{10}}^9\text{J}$ of heat transfer when burned?

Assume that the turbines at a coal-powered power plant were upgraded, resulting in an improvement in efficiency of 3.32%. Assume that prior to the upgrade the power station had an efficiency of 36% and that the heat transfer into the engine in one day is still the same at $2\text{.}\text{50}\times {\text{10}}^{\text{14}}\text{J}$. (a) How much more electrical energy is produced due to the upgrade? (b) How much less heat transfer occurs to the environment due to the upgrade?

A certain gasoline engine has an efficiency of 30.0%. What would the hot reservoir temperature be for a Carnot engine having that efficiency, if it operates with a cold reservoir temperature of $2\text{00}\text{º}\text{C}$?

(a) What is the hot reservoir temperature of a Carnot engine that has an efficiency of 42.0% and a cold reservoir temperature of $\text{27}\text{.}0\text{º}\text{C}$? (b) What must the hot reservoir temperature be for a real heat engine that achieves 0.700 of the maximum efficiency, but still has an efficiency of 42.0% (and a cold reservoir at $\text{27}\text{.}0\text{º}\text{C}$)? (c) Does your answer imply practical limits to the efficiency of car gasoline engines?

Practical steam engines utilize $\text{450}\text{º}\text{C}$ steam, which is later exhausted at $\text{270}\text{º}\text{C}$. (a) What is the maximum efficiency that such a heat engine can have? (b) Since $\text{270}\text{º}\text{C}$ steam is still quite hot, a second steam engine is sometimes operated using the exhaust of the first. What is the maximum efficiency of the second engine if its exhaust has a temperature of $\text{150}\text{º}\text{C}$? (c) What is the overall efficiency of the two engines? (d) Show that this is the same efficiency as a single Carnot engine operating between $\text{450}\text{º}\text{C}$ and $\text{150}\text{º}\text{C}$. Explicitly show how you follow the steps in the Problem-Solving Strategies for Thermodynamics.

Would you be willing to financially back an inventor who is marketing a device that she claims has 25 kJ of heat transfer at 600 K, has heat transfer to the environment at 300 K, and does 12 kJ of work? Explain your answer.

Unreasonable Results

Calculate the cold reservoir temperature of a steam engine that uses hot steam at $\text{450}\text{º}\text{C}$ and has a Carnot efficiency of 0.700. (b) What is unreasonable about the temperature? (c) Which premise is unreasonable?

What is the coefficient of performance of an ideal heat pump that has heat transfer from a cold temperature of $-\text{25}\text{.}0\text{º}\text{C}$ to a hot temperature of $\text{40}\text{.}0\text{º}\text{C}$?

What is the best coefficient of performance possible for a hypothetical refrigerator that could make liquid nitrogen at $-\text{200}\text{º}\text{C}$ and has heat transfer to the environment at $\text{35}\text{.}0\text{º}\text{C}$?

(a) What is the best coefficient of performance for a heat pump that has a hot reservoir temperature of $\text{50}\text{.}0\text{º}\text{C}$ and a cold reservoir temperature of $-\text{20}\text{.0ºC}$? (b) How much heat transfer occurs into the warm environment if $3\text{.60}\times {\text{10}}^7\text{J}$ of work ($\text{10}\text{.}0\text{kW}\cdot \text{h}$) is put into it? (c) If the cost of this work input is $\text{10.0 cents/kW}\cdot \text{h}$, how does its cost compare with the direct heat transfer achieved by burning natural gas at a cost of 85.0 cents per therm. (A therm is a common unit of energy for natural gas and equals $1\text{.}\text{055}\times {\text{10}}^8\text{J}$.)

Suppose you want to operate an ideal refrigerator with a cold temperature of $-\text{10}\text{.}\mathrm{0º}\text{C}$, and you would like it to have a coefficient of performance of 7.00. What is the hot reservoir temperature for such a refrigerator?

A 4-ton air conditioner removes $5\text{.}\text{06}\times {\text{10}}^7\text{J}$ (48,000 British thermal units) from a cold environment in 1.00 h. (a) What energy input in joules is necessary to do this if the air conditioner has an energy efficiency rating ($\text{EER}$) of 12.0? (b) What is the cost of doing this if the work costs 10.0 cents per $3\text{.}\text{60}\times {\text{10}}^6\text{J}$ (one kilowatt-hour)? (c) Discuss whether this cost seems realistic. Note that the energy efficiency rating ($\text{EER}$) of an air conditioner or refrigerator is defined to be the number of British thermal units of heat transfer from a cold environment per hour divided by the watts of power input.

(a) On a winter day, a certain house loses $5\text{.}\text{00}\times {\text{10}}^8\text{J}$ of heat to the outside (about 500,000 Btu). What is the total change in entropy due to this heat transfer alone, assuming an average indoor temperature of $\text{21.0º C}$ and an average outdoor temperature of $\mathrm{5.00º\; C}$? (b) This large change in entropy implies a large amount of energy has become unavailable to do work. Where do we find more energy when such energy is lost to us?

A hot rock ejected from a volcano’s lava fountain cools from $\text{1100º C}$ to $\text{40.0º C}$, and its entropy decreases by 950 J/K. How much heat transfer occurs from the rock?

The Sun radiates energy at the rate of $3\text{.}\text{80}\times {\text{10}}^{\text{26}}\text{W}$ from its $\text{5500º C}$ surface into dark empty space (a negligible fraction radiates onto Earth and the other planets). The effective temperature of deep space is $-\text{270º C}$. (a) What is the increase in entropy in one day due to this heat transfer? (b) How much work is made unavailable?

What is the decrease in entropy of 25.0 g of water that condenses on a bathroom mirror at a temperature of $\text{35.0º C}$, assuming no change in temperature and given the latent heat of vaporization to be 2450 kJ/kg?

A large electrical power station generates 1000 MW of electricity with an efficiency of 35.0%. (a) Calculate the heat transfer to the power station, $Q_{\text{h}}$, in one day. (b) How much heat transfer $Q_{\text{c}}$ occurs to the environment in one day? (c) If the heat transfer in the cooling towers is from $\text{35.0º C}$ water into the local air mass, which increases in temperature from $\text{18.0º C}$ to $\text{20.0º C}$, what is the total increase in entropy due to this heat transfer? (d) How much energy becomes unavailable to do work because of this increase in entropy, assuming an $\text{18.0º C}$ lowest temperature? (Part of $Q_{\text{c}}$ could be utilized to operate heat engines or for simply heating the surroundings, but it rarely is.)

Using Table 15.4, verify the contention that if you toss 100 coins each second, you can expect to get 100 heads or 100 tails once in $2\times {\text{10}}^{\text{22}}$ years; calculate the time to two-digit accuracy.

(a) If tossing 100 coins, how many ways (microstates) are there to get the three most likely macrostates of 49 heads and 51 tails, 50 heads and 50 tails, and 51 heads and 49 tails? (b) What percent of the total possibilities is this? (Consult Table 15.4.)

(a) What is the change in entropy if you start with 10 coins in the 5 heads and 5 tails macrostate, toss them, and get 2 heads and 8 tails? (b) How much more likely is 5 heads and 5 tails than 2 heads and 8 tails? (Take the ratio of the number of microstates to find out.) (c) If you were betting on 2 heads and 8 tails would you accept odds of 252 to 45? Explain why or why not.

(a) Construct a table showing the macrostates and all of the individual microstates for tossing 6 coins. (Use Table 15.5 as a guide.) (b) How many macrostates are there? (c) What is the total number of microstates? (d) What percent chance is there of tossing 5 heads and 1 tail? (e) How much more likely are you to toss 3 heads and 3 tails than 5 heads and 1 tail? (Take the ratio of the number of microstates to find out.)