Kinetic theory of gas and first law of thermodynamics – problems and solutions

1. Ideal gases are in a container with a volume of 4 liters and its pressure is 3 atm (1 atm = 10^{5} N.m^{-2}). The ideal gases heated at a constant pressure from 27^{o}C to 87^{o}C. The heat capacity of the gas is 9 J.K^{-1}. What is the final volume of gases and the change of internal energy of gases?

Solution

**Isobaric process (constant pressure)**

__Known :__

The initial volume of gas (V_{1}) = 4 liters

The initial temperature of gas (T_{1}) = 27^{o}C + 273 = 300 K

The final temperature of gas (T_{2}) = 87^{o}C + 273 = 360 K

The pressure of gas (P) = 3 atm = 3 x 10^{5} N.m^{-2}

The heat capacity of gas (C) = 9 J.K^{-1}

__Wanted :__ The final volume of gas (V_{2}) and the change of internal energy of gas (ΔU)

__Solution :__

Determine the final volume using the equation of Charles’s law (isobaric process or constant pressure) :

__The change in volume :__

1 liter = 0.001 m^{3 }

Initial volume (V_{1}) = 4 (0.001 m^{3}) = 0.004 m^{3}

Final volume (V_{2}) = 4.8 (0.001 m^{3}) = 0.0048 m^{3}

The change in volume (ΔV) = V_{2} – V_{1 }= 0.0048 m^{3} – 0.004 m^{3} = 0.008 m^{3}.

__The change in temperature :__

The change in temperature (ΔT) = T_{2} – T_{1} = 360 K – 300 K = 60 K

Determine the change of the internal energy (ΔU) of the ideal gas using the equation of the first law of thermodynamics.

ΔU = Q – W

*ΔU = the change of the internal energy, Q = heat, W = work*

Determine work (W) at constant pressure :

W = P ΔV = (3 x 10^{5})(0.0008) = (3 x 10^{1})(8) = (30)(8) = 240 Joule

Determine heat (Q) using the equation of the heat capacity (C) :

C = Q / ΔT

Q = (C)(ΔT) = (9)(60) = 540 Joule

Determine the change of the internal energy :

ΔU = Q – W = 540 Joule – 240 Joule = 300 Joule.

2. 6 liters of ideal gases at 2 atm are in a container (1 atm = 10^{5} N.m^{-2}). The gas heated from 27^{o}C to 77^{o}C at a constant pressure. If the heat capacity of gas is 5 J.K^{-1}, what is the final volume and the change of internal energy of the gas.

Solution :

**Isobaric process (constant pressure)**

__Known :__

The initial volume of the ideal gases (V_{1}) = 6 liters

The initial temperature of the ideal gases (T_{1}) = 27^{o}C + 273 = 300 K

The final temperature of the ideal gases (T_{2}) = 77^{o}C + 273 = 350 K

The pressure of the ideal gases (P) = 2 atm = 2 x 10^{5} N.m^{-2}

The heat capacity of gases (C) = 5 J.K^{-1}

__Wanted:__ The final volume of the gas (V_{2}) and the change of internal energy of the gas (ΔU)

__Solution :__

Determine the final volume of gas using the equation of Charles’s law (isobaric process or constant pressure) :

__The change in volume :__

1 liter = 0.001 m^{3 }

The initial volume (V_{1}) = 6 (0.001 m^{3}) = 0.006 m^{3}

The final volume (V_{2}) = 7 (0.001 m^{3}) = 0.007 m^{3}

The change in volume (ΔV) = V_{2} – V_{1} = 0.007 m^{3} – 0.006 m^{3} = 0.001 m^{3}

__The change in temperature :__

The change in temperature (ΔT) = T_{2} – T_{1 }= 350 K – 300 K = 50 K

Determine the change of the internal energy (ΔU) of the ideal gases using the equation of the first law of thermodynamics.

ΔU = Q – W

*ΔU = the change in the internal energy, Q = heat, W = work*

Determine work (W) at constant pressure :

W = P ΔV = (2 x 10^{5})(0.001) = (2 x 10^{2})(1) = (200)(1) = 200 Joule

Determine heat (Q) using the equation of the heat capacity (C) :

C = Q / ΔT

Q = (C)(ΔT) = (5)(50) = 250 Joule

Determine the change in the internal energy :

ΔU = Q – W = 250 Joule – 200 Joule = 50 Joule.

**What are the main assumptions of the kinetic theory of gases?****Answer:**The kinetic theory of gases assumes that: (a) Gases consist of a large number of tiny particles that are in constant random motion; (b) These particles are far apart relative to their size; (c) Collisions between gas particles, or between a particle and the walls of its container, are elastic (i.e., no kinetic energy is lost); (d) There are no intermolecular forces between gas particles; and (e) The average kinetic energy of the gas particles is directly proportional to the absolute temperature of the gas.

**How is the average kinetic energy of gas molecules related to the temperature of the gas?****Answer:**The average kinetic energy of gas molecules is directly proportional to the absolute temperature (in Kelvin) of the gas. As the temperature increases, the average kinetic energy of the molecules also increases.

**What is the significance of absolute zero in terms of molecular motion?****Answer:**Absolute zero (0 Kelvin) is theoretically the temperature at which all molecular motion stops. It represents the lowest possible temperature where nothing could be colder and no heat energy remains in a substance.

**What is the first law of thermodynamics in terms of internal energy, heat, and work?****Answer:**The first law of thermodynamics states that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system on its surroundings: ΔU = Q – W.

**In an isochoric process, what happens to the work done by or on the system?****Answer:**In an isochoric process, the volume remains constant. Hence, there is no work done by or on the system, as work in such cases is given by the pressure-volume work, which would be zero if the volume doesn’t change.

**What does it mean when we say that a gas is “ideal”?****Answer:**An “ideal” gas is one that follows the ideal gas law (PV = nRT) under all conditions of temperature and pressure. It also means that the gas follows the postulates of the kinetic theory perfectly, with no intermolecular forces and perfectly elastic collisions.

**How do real gases deviate from the behavior of ideal gases?****Answer:**Real gases deviate from ideal behavior at high pressures and low temperatures. This is due to the presence of intermolecular forces and the finite size of gas molecules. The deviations are described by the van der Waals equation and other similar equations.

**Why can the first law of thermodynamics be thought of as a conservation law?****Answer:**The first law of thermodynamics can be considered a conservation law because it states that energy cannot be created or destroyed, only transferred or converted from one form to another. This mirrors the principle of conservation of energy.