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Chapter 6: Problem 100

Calculate the internal energy of a Goodyear blimp filled with helium gas at \(1.2 \times 10^{5} \mathrm{~Pa}\). The volume of the blimp is \(5.5 \times 10^{3} \mathrm{~m}^{3} .\) If all the energy were used to heat 10.0 tons of copper at \(21^{\circ} \mathrm{C},\) calculate the final temperature of the metal. (Hint: See Section 5.6 for help in calculating the internal energy of a gas. 1 ton \(\left.=9.072 \times 10^{5} \mathrm{~g} .\right)\)

Short Answer

Expert verified
The final temperature of the copper is approximately 557°C.

Step by step solution

01

Calculate the Internal Energy of the helium gas

Use the formula for the internal energy of an ideal gas, which is given by \( U = \frac{3}{2} nRT \), where n is the number of moles and R is the Universal gas constant. For a gas such as helium, at standard atmospheric pressure and temperature, we use n= \( \frac{PV}{RT} \), so the energy will be \( U = \frac{3}{2} P V \), where P is the pressure of the gas and V is the volume.
02

Compute the Internal Energy

Substitute the given values to calculate the energy. \( U = \frac{3}{2} \times 1.2 \times 10^{5} \times 5.5 \times 10^{3} = 9.9 \times 10^{8} \) Joules
03

Write the Formula for Heat Transfer for Copper

The formula for heat transfer is \( q= mc \Delta T \), where m is mass, c is specific heat capacity and \( \Delta T \) is the change in temperature.
04

Substitute the Known Values

The energy gained by the Copper is equal to the energy lost by the helium gas. The mass of the copper, m is given as 10 tons, which is \( 10 \times 9.072 \times 10^{5} \) g. The specific heat of copper, c, is 0.385 J/g°C and the initial temperature, \( T_1 \), is 21°C. Substitute these values into the equation and solve for \( T_2 \), the final temperature. So the equation becomes: \( 9.9 \times 10^{8} = 10 \times 9.072 \times 10^{5} \times 0.385 \times (T_2 - 21) \)
05

Solve for the Final Temperature

Solving for \( T_2 \) gives the final temperature of the Copper as approximately 557°C

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Most popular questions from this chapter

The average temperature in deserts is high during the day but quite cool at night, whereas that in regions along the coastline is more moderate. Explain.

Construct a table with the headings \(q, w, \Delta E,\) and \(\Delta H .\) For each of the following processes, deduce whether each of the quantities listed is positive \((+)\) negative \((-),\) or zero (0) . (a) Freezing of benzene. (b) Compression of an ideal gas at constant temperature. (c) Reaction of sodium with water. (d) Boiling liquid ammonia. (e) Heating a gas at constant volume. (f) Melting of ice.

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I he enthaipy of combustion benzo1c acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\) is commonly used as the standard for calibrating constant-volume bomb calorimeters; its value has been accurately determined to be \(-3226.7 \mathrm{~kJ} / \mathrm{mol}\). When \(1.9862 \mathrm{~g}\) of benzoic acid are burned in a calorimeter, the temperature rises from \(21.84^{\circ} \mathrm{C}\) to \(25.67^{\circ} \mathrm{C}\). What is the heat capacity of the bomb? (Assume that the quantity of water surrounding the bomb is exactly \(2000 \mathrm{~g} .)\)
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