Question

Given each of the following sets of values for three of the gas variables, calculate the unknown quantity. a. \(P=1.034\) atm \(; V=21.2 \mathrm{~mL} ; n=0.00432 \mathrm{~mol} ; T=? \mathrm{~K}\) b. \(P=?\) atm \(; V=1.73 \mathrm{~mL} ; n=0.000115 \mathrm{~mol} ; T=182 \mathrm{~K}\) c. \(P=1.23 \mathrm{~mm} \mathrm{Hg} ; V=? \mathrm{~L} ; n=0.773 \mathrm{~mol} ; T=152 ? \mathrm{C}\)

Short answer

a. For the given values of $P=1.034 \mathrm{~atm}$, $V=21.2 \mathrm{~mL}$, and $n=0.00432 \mathrm{~mol}$, we find the unknown temperature $T=70.15 \mathrm{~K}$. b. For the given values of $V=1.73 \mathrm{~mL}$, $n=0.000115 \mathrm{~mol}$, and $T=182 \mathrm{~K}$, we find the unknown pressure $P=6.12 \mathrm{~atm}$. c. For the given values of $P=1.23 \mathrm{~mm Hg}$, $n=0.773 \mathrm{~mol}$, and $T=152 \mathrm{~°C}$, we find the unknown volume $V=20.74 \mathrm{~L}$.

Step-by-step solution

Convert volume to liters

We need to convert volume from mL to L: \(V = 21.2 \mathrm{~mL} \times \frac{1 \mathrm{~L}}{1000 \mathrm{~mL}} = 0.0212 \mathrm{~L} \)

Solve for temperature T using Ideal Gas Law

Now we can use the Ideal Gas Law equation to solve for the unknown temperature: \(PV = nRT\) \(T = \frac{PV}{nR}\) Given: P = 1.034 atm; V = 0.0212 L; n = 0.00432 mol; and R = 0.0821 L⋅atm/mol⋅K, \(T = \frac{(1.034 \mathrm{~atm})(0.0212 \mathrm{~L})}{(0.00432 \mathrm{~mol})(0.0821 \mathrm{~L⋅atm/mol⋅K})} = 70.15 \mathrm{~K}\) Answer: T = 70.15 K Case b. P = ? atm; V = 1.73 mL; n = 0.000115 mol; T = 182 K

Convert volume to liters

We need to convert volume from mL to L: \(V = 1.73 \mathrm{~mL} \times \frac{1 \mathrm{~L}}{1000 \mathrm{~mL}} = 0.00173 \mathrm{~L} \)

Solve for pressure P using Ideal Gas Law

Now we can use the Ideal Gas Law equation to solve for the unknown pressure: \(PV = nRT\) \(P = \frac{nRT}{V}\) Given: V = 0.00173 L; n = 0.000115 mol; T = 182 K; and R = 0.0821 L⋅atm/mol⋅K, \(P = \frac{(0.000115 \mathrm{~mol})(0.0821 \mathrm{~L⋅atm/mol⋅K})(182 \mathrm{~K})}{0.00173 \mathrm{~L}} = 6.12 \mathrm{~atm}\) Answer: P = 6.12 atm Case c. P = 1.23 mm Hg; V = ? L; n = 0.773 mol; T = 152 °C

Convert temperature to Kelvin and pressure to atm

We need to convert temperature from Celsius to Kelvin: \(T = 152 \mathrm{~°C} + 273.15 = 425.15 \mathrm{~K}\) We need to convert pressure from mm Hg to atm: \(P = 1.23 \mathrm{~mmHg} \times \frac{1 \mathrm{~atm}}{760 \mathrm{~mmHg}} = 0.00162 \mathrm{~atm} \)

Solve for volume V using Ideal Gas Law

Now we can use the Ideal Gas Law equation to solve for the unknown volume: \(PV = nRT\) \(V = \frac{nRT}{P}\) Given: P = 0.00162 atm; n = 0.773 mol; T = 425.15 K; and R = 0.0821 L⋅atm/mol⋅K, \(V = \frac{(0.773 \mathrm{~mol})(0.0821 \mathrm{~L⋅atm/mol⋅K})(425.15 \mathrm{~K})}{0.00162 \mathrm{~atm}} = 20.74 \mathrm{~L}\) Answer: V = 20.74 L

Key concepts

Gas Variables

Understanding gas variables is fundamental to solving problems related to gases. These variables include pressure (P), volume (V), moles (n), and temperature (T), and they are interconnected through the ideal gas law.

Pressure, often measured in atmospheres (atm) or millimeters of mercury (mmHg), is the force the gas exerts on the container. Volume is the space occupied by a gas, usually in liters (L) or milliliters (mL). Moles (n) represent the amount of substance, which directly affects the number of gas particles in a given volume. Temperature (T) is measured in Kelvins (K) and reflects the average kinetic energy of gas particles; it's essential to convert from Celsius to Kelvin, adding 273.15 to the Celsius temperature, since Kelvin is the SI unit for temperature and provides an absolute scale.

These gas variables are not isolated; a change in one often leads to a change in others, as dictated by the ideal gas equation.

Solving Gas Laws

To solve gas law problems, it is important to follow a systematic approach:

• Identify the given gas variables and what needs to be determined.
• Convert units if necessary, such as volume to liters or pressure to atmospheres, for consistency with the gas constant (R).
• Apply the correct equation; for ideal gas problems, this is typically the ideal gas law.
• Isolate the variable you need to find and plug in the known values.
• Carry out the math to solve for the unknown variable.

Complicated as it may seem, comprehending the relationships between these variables allows for substitution into the ideal gas law formula. This method lets you solve for any unknown variable as long as the other three are known, aligning with our goal for students to easily grasp all concepts.

Ideal Gas Equation

The ideal gas equation is a cornerstone of solving gas-related problems and is expressed as: \[ PV = nRT \] where P represents pressure, V is volume, n is the number of moles, T is the absolute temperature in Kelvins, and R is the ideal gas constant (0.0821 L⋅atm/mol⋅K).

This equation assumes that the gas particles are point particles that move randomly and do not interact, which is a good approximation under many conditions. The ideal gas law can be used to find any one of the gas variables if the other three are known, offering a reliable method for examining the behavior of gases under various conditions. Whether you're calculating the pressure of a balloon or determining the amount of gas needed to inflate it, utilizing the ideal gas law becomes a practical tool for many scientific and real-world applications.