22. The First Law of Thermodynamics

2.0 mol of gas are at 30 °C and a pressure of 1.5 atm. How much work must be done on the gas to compress it to one third of its initial volume at constant pressure?

During an isothermal compression of an ideal gas, $410$ J of heat must be removed from the gas to maintain constant temperature. How much work is done by the gas during the process?

On a warm summer day, a large mass of air (atmospheric pressure $1.01\times {10}^5$ Pa) is heated by the ground to $26.0$°C and then begins to rise through the cooler surrounding air. (This can be treated approximately as an adiabatic process; why?) Calculate the temperature of the air mass when it has risen to a level at which atmospheric pressure is only $0.850\times {10}^5$ Pa. Assume that air is an ideal gas, with $g=1.40$. (This rate of cooling for dry, rising air, corresponding to roughly $1$ C° per $100$ m of altitude, is called the dry adiabatic lapse rate.)

FIGURE CP19.80 shows a thermodynamic process followed by 0.015 mol of hydrogen. How much heat energy is transferred to the gas?

A cylinder contains $0.0100$ mol of helium at $T = 27.0$°C. What accounts for the difference between your answers to parts (a) and (b)? In which case is more heat required? What becomes of the additional heat?

(a) How much heat is needed to raise the temperature to $67.0$°C while keeping the volume constant? Draw a $pV$-diagram for this process.

(b) If instead the pressure of the helium is kept constant, how much heat is needed to raise the temperature from $27.0$°C to $67.0$°C? Draw a $pV$-diagram for this process.

(II) A 2.00-mole sample of N₂ gas at 0°C is heated to 150°C at constant pressure (1.00 atm). Determine (a) the change in internal energy, (b) the work the gas does, and (c) the heat added to it.

The volume of a gas is halved during an adiabatic compression that increases the pressure by a factor of 2.5. What is the specific heat ratio γ?

An experimenter adds $970$ J of heat to $1.75$ mol of an ideal gas to heat it from $10.0$°C to $25.0$°C at constant pressure. The gas does $+223$ J of work during the expansion. Calculate $\gamma$ for the gas.

An ideal-gas process is described by p=cV^1/2, where c is a constant. Find an expression for the work done on the gas in this process as the volume changes from V₁ to V₂.

A container holds 1.0 g of oxygen at a pressure of 8.0 atm. How much will the temperature increase if this amount of heat energy is transferred to the gas at constant volume?

At very low temperatures, the molar specific heat of many substances varies as the cube of the absolute temperature: C = k (T³ / T³₀) which is sometimes called Debye’s law. For rock salt, T₀ = 281 K and k = 1940 J/mol · K. Determine the heat needed to raise 2.75 mol of salt from 22.0 K to 46.0 K.

A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. How much heat energy is transferred, and in which direction?

(II) Show that the work done by n moles of an ideal gas when it expands adiabatically is W = nC_v( T₁ - T₂) , where T₁ and T₂ are the initial and final temperatures, and C_v is the molar specific heat at constant volume.

5.0 g of nitrogen gas at 20°C and an initial pressure of 3.0 atm undergo an isobaric expansion until the volume has tripled. How much heat energy is transferred to the gas to cause this expansion?

A monatomic ideal gas that is initially at $1.50\(\times\)10^5$ Pa and has a volume of $0.0800$ m³ is compressed adiabatically to a volume of $0.0400$ m³. What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?