Conceptual Questions (5 points each)

Questions 1 -- 6 refer to the following situation:

Mystery substance X has the following thermodynamic data:

1. How much heat is required to heat 1 kg of liquid X from 10 C to 210 C?

   (c) 100 kcal

   To heat liquid X through C requires

   

2. How much heat is required to boil 1 kg of liquid X at 210 C?

   (d) 150 kcal

   To boil liquid X at 210 C requires

   

   Note that , the heat of vaporization, applies to boiling the liquid; not , the heat of fusion, which applies to the melting/freezing transition.

3. What is the fractional change in its volume, when liquid X decreases in temperature by 10 C?

   (d) Liquid X's volume decreases by 4%.

   The fractional change in volume, for C, is

   

   for a decrease of 4%.

4. A sample of liquid X at 80 C sits in a 0 C room, and cools by radiation only. It loses energy at the rate 1 W. If the room temperature were increased to 40 C, at what rate would the 80 C sample of liquid X now lose energy?

   (c) 0.6 W

   The sample loses radiative energy because it emits more energy than it absorbs. Specifically, it loses energy at the rate

   

   where all temperatures are expressed in Kelvin. When the room temperature is increased to C, this rate changes because is altered by a factor

   

   Since the old rate of energy loss, , is given as 1 W, the rate in the warmer room must be 0.6 W.

   
   

   Questions 5 and 6 pertain to the following specific process undergone by substance X:

   
   

   1 kg of substance X is heated from an initial temperature of -10 C to a final state which is a mixture of 0.3 kg of liquid X and 0.70 kg of gas X.

   
   

5. Which one of the following is an accurate qualitative depiction of the sample's evolution in pressure and temperature?

   

6. Which one of the following is an accurate qualitative depiction of the sample's temperature variation as heat is added?

   

   Questions 7 through 12 pertain to the situation described below:

   An ideal monatomic gas originally in state A at temperature 27 C can be taken to state B at temperature via any of the thermodynamic processes shown.

   
   For these questions you may choose to use the ideal gas constants , though they are not strictly necessary.

7. Which of the following is an accurate ordering of the amount of work done (W) by the gas in each of these processes?

   (a)

   Work done by the gas is given by the area under the PV curve. By inspection (for example, counting up full boxes under the curves), this is greatest for I, then smaller for II, then III, then IV.

8. How much work is done by the gas in process I? (Note the conversion factor 1 atm l = 100 J.)

   (b) 600 J

   Process I occurs at constant pressure, so the work done by the gas is just

   

   which, given the conversion factor, is 600 J.

9. What is the temperature of the gas in state B?

   (d) 327 C

   This can be found from the ideal gas law, for constant n, where all temperatures must be expressed in Kelvin:

   

10. In process III, the gas expands from state A to state C isothermally, doing 480 J of work. What is the change in entropy ( ) for the gas during the isothermal process AC?

    (b) 1.6 J/K

    Since the ideal gas expands isothermally, and Q= W = 480 J. is just defined as Q/T, where T is the temperature of the constant temperature process, expressed in Kelvin. This is given as , so

    

    
    

    Questions 11 and 12 both refer to process IV, where the gas expands from state A to state D adiabatically, with C.

    
    

11. What is the change in internal energy of the gas during the adiabatic process AD?

    (a) -225 J

    The change in internal energy, for this ideal monatomic gas, is

    

    Here K. But 150 K is , so using the ideal gas law we find

    

    which gives -225 J using the 1 atm l = 100 J conversion factor.

12. How much work does the gas do in process IV (for the entire process, from A to D to B)?

    (b) 225 J

    Since process AD is adiabatic, , telling us the gas does W = 225 J of work from A to D, using our answer to 11. Since process DB is isochoric (constant volume), the gas does no further work from D to B. Thus the gas does a total of 225 J of work in process IV.

    Questions 13 and 14 pertain to the following situation:

    
    

    A heat engine operates between a hot reservoir at 1500 K and a cold reservoir at 500 K. J of heat is removed from the hot reservoir and J of work is performed.

13. What is the actual efficiency of this engine?

    (a) 0.15

    The engine has efficiency

    

14. What is the ideal (Carnot) efficiency of this engine?

    (d) 0.67

    The Carnot efficiency of the engine is

    

Quantitative Problems (Point Value as Marked)

1. (30 points)
   
   A 350 g copper ingot at C is placed in a calorimeter with 420 g of water at C. Specific heats for copper, water, and ice are

   

   while the heat of fusion for water is .

   1. How much heat would be required to raise the temperature of the copper ingot to C?
   2. What is the final state of the system? Give both the final temperature, and the masses of water and/or ice present.

   
   
   
   
   
   
   

   1. For such heating, the copper must gain

      

   2. We use our answer from (a), stating that the copper must gain to raise its temperature to C, to help us assess what the final temperature must be. For in this final state, the heat gained by the copper must equal the heat lost by the water.

      First, were the water to cool to C, it would give up

      

      This is not enough to heat the copper to the same temperature, C, so the water must continue to give up heat.

      Next, were the water to fully freeze at C, it would give up an additional

      

      Together with the heat lost by the water in cooling, this is more than enough to heat the copper to the same final temperature, C. Thus we know that the final temperature must be C, with only a portion of the water frozen to ice so that, in reaching the final state, .

      Setting up this equality,

      

      using our partial results above. Thus

      

      and

      

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