Exercises: Ideal gases

In a laboratory, the behaviour of a gas enclosed in a cylinder with a piston is studied. The state of the gas is described by four variables: pressure, volume, temperature and amount of substance. The diagram shows the gas enclosed in the cylinder with its four variables.

1. a) Explain in your own words what each of these variables indicates.
2. b) State the SI unit for each one.
3. c) When applying the ideal gas equation, $p\times V=n\times R\times T$, with $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$, in which units should pressure, volume, temperature and amount of substance be expressed?

A rigid laboratory container contains a gas at $2\ \text{atm}$ and $27\ \celsius$, and occupies a volume of $24.6\ \text{L}$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Express the temperature in kelvin.
2. b) Apply the ideal gas equation, $p\times V=n\times R\times T$, and calculate how many moles of gas are in the container.

A cylinder with a piston contains $2\ \text{mol}$ of nitrogen at $300\ \text{K}$, subjected to a pressure of $4\ \text{atm}$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Write down the ideal gas equation and rearrange for volume.
2. b) Calculate the volume occupied by the gas.

In an experiment, exactly $1\ \text{mol}$ of an ideal gas is enclosed in a $22.4\ \text{L}$ container kept at $273\ \text{K}$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Calculate the pressure of the gas, in atmospheres.
2. b) Which known value does the result approximate? What are these pressure and temperature conditions called?

A small weather balloon contains $2\ \text{mol}$ of helium. At a certain point in its ascent, the sensors record a pressure of $2\ \text{atm}$ and a volume of $24.6\ \text{L}$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Calculate the temperature of the helium in kelvin.
2. b) Express it in degrees Celsius.

In chemistry, standard temperature and pressure (STP) are used as reference conditions.

1. a) Which values of pressure and temperature define standard conditions (STP)? Express the temperature in kelvin as well.
2. b) Use the ideal gas equation, $p\times V=n\times R\times T$ with $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$, to calculate the volume occupied by $1\ \text{mol}$ of any ideal gas at standard temperature and pressure.
3. c) What is this reference volume called?

In a laboratory, $32\ \text{g}$ of oxygen, $\text{O}_2$, is weighed. Data: molar mass of $\text{O}_2$, $32\ \text{g/mol}$; molar volume at standard temperature and pressure, $22.4\ \text{L/mol}$.

1. a) How many moles of oxygen is this?
2. b) What volume would this gas occupy at standard temperature and pressure?

A $5\ \text{L}$ cylinder contains $10\ \text{g}$ of nitrogen gas. The figure compares this cylinder with the same volume of water using the particle model.

1. a) Calculate the density of the gas in $\text{g/L}$ (mass divided by volume).
2. b) The density of water is $1000\ \text{g/L}$: how many times denser is water than this gas?
3. c) Explain, using the particle model, why gases have such low densities compared to liquids.

A $\text{CO}_2$ fire extinguisher has a container of $24.6\ \text{L}$ and stores the gas at $10\ \text{atm}$ and $27\ \celsius$. Data: molar mass of $\text{CO}_2$, $44\ \text{g/mol}$; $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Express the temperature in kelvin.
2. b) Calculate the moles of $\text{CO}_2$ contained in the fire extinguisher.
3. c) What mass of $\text{CO}_2$ is inside?

$\text{CO}_2$ is denser than air, which explains why it can accumulate at the bottom of wells and poorly ventilated cellars. Data: molar mass of $\text{CO}_2$, $44\ \text{g/mol}$; $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$; density of air at STP, $1.29\ \text{g/L}$.

1. a) Calculate the mass of $1\ \text{mol}$ of $\text{CO}_2$ and the volume it occupies at STP ($1\ \text{atm}$ and $273\ \text{K}$).
2. b) Calculate the density of $\text{CO}_2$ at STP, to one decimal place.
3. c) Compare it with that of air and explain why $\text{CO}_2$ accumulates in low areas.

A balloon contains $0.25\ \text{mol}$ of helium at $27\ \celsius$ and the barometer measures an external pressure of $380\ \text{mmHg}$. Data: $1\ \text{atm}=760\ \text{mmHg}$; $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Express the temperature in kelvin and the pressure in atmospheres.
2. b) Calculate the volume of the balloon.

A diving cylinder of $12.3\ \text{L}$ is filled with air at $200\ \text{atm}$ and $27\ \celsius$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$. The pressure-volume graph shows the compressed air in the cylinder and its expansion at constant temperature to surface pressure.

1. a) Express the temperature in kelvin and calculate how many moles of air the cylinder contains.
2. b) Calculate what volume all that air would occupy at surface pressure, $1\ \text{atm}$, and at the same temperature.
3. c) A diver consumes about $60\ \text{L}$ of air per minute measured at the surface: estimate, in minutes, the theoretical duration of the cylinder.

A cylinder containing an unknown gas is received in a laboratory. Measuring its mass and volume gives a density of $1.43\ \text{g/L}$ at STP. Data: molar volume at STP, $22.4\ \text{L/mol}$.

1. a) Calculate the mass of $22.4\ \text{L}$ of the gas, i.e. its molar mass.
2. b) Which gas could it be: $\text{H}_2$ ($2\ \text{g/mol}$), $\text{O}_2$ ($32\ \text{g/mol}$) or $\text{CO}_2$ ($44\ \text{g/mol}$)? Justify your answer.

A factory boiler contains $4\ \text{mol}$ of water vapour in a volume of $16.4\ \text{L}$ at a pressure of $10\ \text{atm}$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Calculate the temperature of the vapour in kelvin.
2. b) Express it in degrees Celsius.
3. c) Explain why it is essential to use kelvin and not degrees Celsius in the ideal gas equation: what would happen to the result if you used the temperature in $\celsius$?

A balloon at a fair contains $2\ \text{mol}$ of helium at $1\ \text{atm}$ and $300\ \text{K}$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$; $N_A=6.022\times10^{23}\ \text{mol}^{-1}$.

1. a) Calculate the volume of the balloon.
2. b) How many helium atoms does it contain? Express the result in standard form, to two decimal places.

Two identical $11.2\ \text{L}$ containers are both at STP ($1\ \text{atm}$ and $273\ \text{K}$): one contains nitrogen, $\text{N}_2$, and the other hydrogen, $\text{H}_2$. Data: molar masses of $\text{N}_2$ and $\text{H}_2$, $28\ \text{g/mol}$ and $2\ \text{g/mol}$; molar volume at STP, $22.4\ \text{L/mol}$.

1. a) How many moles of gas are in each container?
2. b) Calculate the mass of gas in each one.
3. c) Which of the two containers weighs more? Does it therefore contain more particles? Give reasons for your answers.

A hot-air balloon is open at the bottom, so that the pressure of the air inside is always atmospheric pressure, $1\ \text{atm}$. The outside air is at $27\ \celsius$ and the burner heats the inside air up to $127\ \celsius$. Data: average molar mass of air, $29\ \text{g/mol}$; $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$. The diagram represents the balloon and the inside and outside air.

1. a) Express both temperatures in kelvin and calculate the density of the outside air.
2. b) Calculate the density of the hot air inside.
3. c) Compare the mass of $1000\ \text{L}$ ($1\ \text{m}^3$) of air inside and outside, and use this difference to explain why the balloon rises.

A weather balloon is filled on the ground with $100\ \text{mol}$ of helium, at $1\ \text{atm}$ and $300\ \text{K}$. It rises to the stratosphere, where the pressure is $0.2\ \text{atm}$ and the temperature is $-48\ \celsius$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$. The figure shows the balloon at launch and in the stratosphere.

1. a) Calculate the volume of the balloon at the moment of launch and express it in $\text{m}^3$.
2. b) Express the temperature in the stratosphere in kelvin and calculate the volume of the balloon at that altitude.
3. c) In light of the results, explain why these balloons are launched partially inflated and why they end up bursting at high altitude.

At an airport, nitrogen for inflating aircraft tyres is stored in a $24.6\ \text{L}$ cylinder at $100\ \text{atm}$ and $27\ \celsius$. Data: molar mass of $\text{N}_2$, $28\ \text{g/mol}$; $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Calculate the moles of $\text{N}_2$ contained in the cylinder.
2. b) What mass of nitrogen does it store?
3. c) After a working day, the pressure has dropped to $25\ \text{atm}$ at the same temperature. How many moles remain? What mass of nitrogen has been used?

An illuminated sign contains a pure noble gas. In the laboratory, its density is measured: $1.79\ \text{g/L}$ at $2\ \text{atm}$ and $273\ \text{K}$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Calculate the volume that $1\ \text{mol}$ of the gas would occupy at $2\ \text{atm}$ and $273\ \text{K}$.
2. b) Use the density to calculate the mass of that volume, i.e. the molar mass of the gas.
3. c) Identify it among the noble gases: He ($4\ \text{g/mol}$), Ne ($20\ \text{g/mol}$), Ar ($40\ \text{g/mol}$) and Kr ($84\ \text{g/mol}$).
4. d) Give reasons, using the ideal gas equation, why increasing the pressure increases the density of a gas even though its molar mass does not change.

A laboratory autoclave with a capacity of $20\ \text{L}$ contains air at $1\ \text{atm}$ and $27\ \celsius$. It is sealed hermetically and heated to $127\ \celsius$. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Calculate the moles of air inside.
2. b) The volume barely changes on heating: calculate the final pressure assuming the same number of moles and the same volume, at $127\ \celsius$.
3. c) Check that the pressure has been multiplied by the same factor as the temperature in kelvin and write down this proportional relationship.
4. d) Explain, using the kinetic model, why pressure increases on heating and why it is dangerous to open the autoclave while hot.

In a fertiliser plant, a $820\ \text{L}$ reactor contains nitrogen, $\text{N}_2$, at $40\ \text{atm}$ and $400\ \text{K}$. Data: molar mass of $\text{N}_2$, $28\ \text{g/mol}$; $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Calculate the moles of $\text{N}_2$ in the reactor.
2. b) What mass of nitrogen do they represent, in kilograms?
3. c) In the process, $600\ \text{mol}$ of $\text{N}_2$ is consumed. If the volume and temperature do not change, calculate the final pressure in the reactor.
4. d) Explain why the pressure drops in the same proportion as the number of moles and which quantity remains constant in the ratio $\frac{p\times V}{n\times T}$.

Two rigid containers, of $10\ \text{L}$ and $30\ \text{L}$, are connected by a tap that is closed. The first contains oxygen at $4\ \text{atm}$ and $300\ \text{K}$; the second is empty. Data: $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$. The diagram shows the setup with the tap still closed.

1. a) Calculate the moles of enclosed oxygen.
2. b) The tap is opened and the gas spreads throughout the entire available volume, without the temperature changing. Calculate the final pressure.
3. c) Explain, using the kinetic model, why the pressure drops even though not a single particle has left the system.

At the summit of Mount Everest, the atmosphere is much less dense than at sea level. Data: average molar mass of air, $29\ \text{g/mol}$; $R=0.082\ \text{atm}\times\text{L}/(\text{mol}\times\text{K})$.

1. a) Calculate the density of air at sea level, at $1\ \text{atm}$ and $15\ \celsius$ ($288\ \text{K}$).
2. b) At the summit, the pressure is $0.33\ \text{atm}$ and the temperature is $-30\ \celsius$ ($243\ \text{K}$): calculate the air density there.
3. c) Calculate how many grams of air a climber inhales in each breath of $2\ \text{L}$ at each of the two locations.
4. d) Use the results to explain why many climbers need oxygen cylinders at the summit even though the proportion of oxygen in the air is the same at both locations.