Temperature, Kinetic Theory, and the Gas Laws

(a) The density of water at 0ºC is very nearly 1000 kg/m³ (it is actually 999.84 kg/m³ ), whereas the density of ice at 0ºC is 917 kg/m³. Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.) (b) What are the implications of this result for biological cells that are frozen?

Show that β ≈ 3α, by calculating the change in volume ΔV of a cube with sides of length L.

Convert an absolute pressure of 7 x 10⁵ N/M² to gauge pressure in lb/in²  (This value was stated to be just less than 90.0 lb/in² in Example 13.9. Is it?)

Confirm that the units of nRT are those of energy for each value of R : (a) 8.31 J/mol K, (b) 1.99 cal/mol K, and (c) 0.0821 L atm/mol K.

Question: (a) The density of water at 0ºC is very nearly (it is actually ), whereas the density of ice at 0ºC is . Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.) (b) What are the implications of this result for biological cells that are frozen?

Question: Show that β ≈ 3α, by calculating the change in volume ΔV of a cube with sides of length L.

Question: Convert an absolute pressure of to gauge pressure in (This value was stated to be just less than in Example 13.9. Is it?)

Question: Confirm that the units of nRT are those of energy for each value of R : (a) 8.31 J/mol K, (b) 1.99 cal/mol K, and (c) 0.0821 L atm/mol K.

A bicycle tire has a pressure of $\mathbf{7}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{ }{\mathbf{\text{N/m}}}^{\mathbf{2}}$ at a temperature of $\mathbf{18}\mathbf{.}\mathbf{0}{\mathbf{ }}^{\mathbf{0}}\mathbf{C}$and contains $\mathbf{\text{2.00 L}}$ of gas. What will its pressure be if you let out an amount of air that has a volume of ${\mathbf{\text{100 cm}}}^{\mathbf{3}}$ at atmospheric pressure? Assume tire temperature and volume remain constant.

A bicycle tire has a pressure of 7.00 x 10⁵ N/m² at a temperature of $\mathbf{18}\mathbf{.}\mathbf{0}\mathbf{°}\mathbf{C}$ and contains 2.00L of gas. What will its pressure be if you let out an amount of air that has a volume of 100 cm³ at atmospheric pressure? Assume tire temperature and volume remain constant.

Question: Suppose that the average velocity (v_rms) of carbon dioxide molecules (molecular mass is equal to 44.0 g/mol) in a flame is found to be 1.05×10⁵ m/s. What temperature does this represent?

Question: Dry air is 78.1% nitrogen. What is the partial pressure of nitrogen when the atmospheric pressure is 1.01×10⁵ N/m²?

At a spot in the high Andes, water boils at 80.0ºC, greatly reducing the cooking speed of potatoes, for example. What is atmospheric pressure at this location?

Question: What is the density of water vapor in g/m³ on a hot dry day in the desert when the temperature is 40.0ºC and the relative humidity is 6.00%?

Question: A deep-sea diver should breathe a gas mixture that has the same oxygen partial pressure as at sea level, where dry air contains 20.9% oxygen and has a total pressure of 1.01×10⁵ N/m². (a) What is the partial pressure of oxygen at sea level? (b) If the diver breathes a gas mixture at a pressure of 2.00×10⁶ N/m², what percent oxygen should it be to have the same oxygen partial pressure as at sea level?

Question: The vapor pressure of water at 40.0ºC is 7.34×10³ N/m². Using the ideal gas law, calculate the density of water vapor in g/m³ that creates a partial pressure equal to this vapor pressure. The result should be the same as the saturation vapor density at that temperature (51.1 g/m³).

Question: Air in human lungs has a temperature of 37.0ºC and a saturation vapor density of 44.0 g/m³. (a) If 2.00 L of air is exhaled and very dry air inhaled, what is the maximum loss of water vapor by the person? (b) Calculate the partial pressure of water vapor having this density, and compare it with the vapor pressure of 6.31×10³ N/m².

Question: If the relative humidity is 90.0% on a muggy summer morning when the temperature is 20.0ºC, what will it be later in the day when the temperature is 30.0ºC, assuming the water vapor density remains constant?

Question: Late on an autumn day, the relative humidity is 45.0% and the temperature is 20.0°C. What will the relative humidity be that evening when the temperature has dropped to 10.0°C, assuming constant water vapor density?

Question: Atmospheric pressure atop Mt. Everest is 3.30×10⁴ N/m². (a) What is the partial pressure of oxygen there if it is 20.9% of the air? (b) What percent oxygen should a mountain climber breathe so that its partial pressure is the same as at sea level, where atmospheric pressure is 1.01×10⁵ N/m²? (c) One of the most severe problems for those climbing very high mountains is the extreme drying of breathing passages. Why does this drying occur?

Question: What is the dew point (the temperature at which 100% relative humidity would occur) on a day when relative humidity is 39.0% at a temperature of 20.0°C?

Question: On a certain day, the temperature is 25.0°C and the relative humidity is 90.0%. How many grams of water must condense out of each cubic meter of air if the temperature falls to 15.0°C? Such a drop in temperature can, thus, produce heavy dew or fog.

Question: The boiling point of water increases with depth because pressure increases with depth. At what depth will freshwater have a boiling point of 150°C, if the surface of the water is at sea level?

Question: (a) At what depth in freshwater is the critical pressure of water reached, given that the surface is at sea level? (b) At what temperature will this water boil? (c) Is a significantly higher temperature needed to boil water at a greater depth?

Question: To get an idea of the small effect that temperature has on Archimedes’ principle, calculate the fraction of a copper block’s weight that is supported by the buoyant force in 0°C water and compare this fraction with the fraction supported in 95.0°C water.

Question: If you want to cook in water at 150°C, you need a pressure cooker that can withstand the necessary pressure. (a) What pressure is required for the boiling point of water to be this high? (b) If the lid of the pressure cooker is a disk 25.0 cm in diameter, what force must it be able to withstand at this pressure?

Question: (a) How many moles per cubic meter of an ideal gas are there at a pressure of 1.00×10¹⁴ N/m² and at 0ºC? (b) What is unreasonable about this result? (c) Which premise or assumption is responsible?

Question: (a) An automobile mechanic claims that an aluminum rod fits loosely into its hole on an aluminum engine block because the engine is hot and the rod is cold. If the hole is 10.0% bigger in diameter than the 22.0ºC rod, at what temperature will the rod be the same size as the hole? (b) What is unreasonable about this temperature? (c) Which premise is responsible?

Question: The temperature inside a supernova explosion is said to be 2.00×10¹³ K. (a) What would the average velocity v_rms of hydrogen atoms be? (b) What is unreasonable about this velocity? (c) Which premise or assumption is responsible?