Question

A steel tank contains $315\mathrm{g}$ of ammonia gas $\left({\mathrm{NH}}_3\right)$ at an absolute pressure of $1.35\times {10}^6\mathrm{Pa}$ and temperature ${77.0}^{\circ }\mathrm{C}.(a)$ What is the volume of the tank? $(b)$ The tank is checked later when the temperature has dropped to ${22.0}^{\circ }\mathrm{C}$ and the absolute pressure has fallen to $8.68\times {10}^5\mathrm{Pa}$. How many grams of gas leaked out of the tank?

Short answer

a) The volume of the tank is calculated to be approximately 39.24 m^3. b) The amount of gas leaked out of the tank is approximately 96.4 grams.

Step-by-step solution

Convert the Temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin. The formula to convert Celsius to Kelvin is T(K) = T(°C) + 273.15. So, the initial temperature, $T_1={77.0}^{\circ }C+273.15=350.15K$ and the final temperature, $T_2={22.0}^{\circ }C+273.15=295.15K$.

Convert Mass into Moles

Use the molar mass of ammonia, which is 17.031 g/mol, to convert the given mass of ammonia into moles. The formula to convert mass to moles is $n=mass/molarmass$. Thus, $n_1=315g/17.031g/mol=18.5mol$.

Determine the Volume of the Tank

Use the ideal gas law equation, PV = nRT, to solve for the volume, $V_1$. Rearrange the equation to $V_1=n_{1}R{T}_1/P_1$. Substitute the given pressure $P_1=1.35x{10}^{6}Pa$, the Gas Constant $R=8.3145J/(molK)$, the initial temperature $T_1=350.15K$ and the initial moles $n_1=18.5mol$.

Find the final moles ($n_2$)

Now using the final conditions, use the ideal gas law to find the final moles of gas. Rearrange the equation to $n_2=P_2{V}_1/(R{T}_2)$. Substitute the final pressure $P_2=8.68x{10}^{5}Pa$, the previously found volume $V_1$, the Gas Constant $R=8.3145J/(molK)$, and the final temperature $T_2=295.15K$.

Calcualte the Difference of Moles

Finally, subtract the final amount of moles from the initial amount to find how many moles of gas leaked out of the tank, $\mathrm{\Delta }n=n_1-n_2$. Then, convert the moles back into grams using the molar mass of ammonia.

Key concepts

Converting Celsius to Kelvin

Understanding temperature conversion is crucial when working with gas law problems. To convert a temperature from Celsius to Kelvin, we add 273.15 to the Celsius value. This step is foundational in gas law problems because measurements need to be in Kelvin to accurately apply the ideal gas law equations. For example, converting an initial temperature of 77.0°C to Kelvin involves the straightforward calculation:

$T(K)={77.0}^{\circ }\mathrm{C}+273.15=350.15K$. Similarly, a final temperature of 22.0°C converts to: $T(K)={22.0}^{\circ }\mathrm{C}+273.15=295.15K$. This conversion ensures that the temperature is on an absolute scale, which directly relates to the kinetic energy of gas particles in ideal gas law calculations.

Molar Mass Calculation

The molar mass of a substance is the weight of one mole of that substance, and it is a pivotal component in relating mass to moles in gas laws. For isotropic substances like ammonia $N{H}_3$, molar mass can be found by summing the atomic masses of its constituent elements from the periodic table. For ammonia, with a molar mass of 17.031 g/mol, we convert the mass of a gas to moles for use in gas law equations by dividing the given mass by the molar mass:

$n=\frac{mass}{molarmass}$, so, $n_1=\frac{315g}{17.031g/mol}=18.5mol$. This conversion is necessary to use the ideal gas law, which requires the amount of gas to be expressed in moles.

Gas Law Equations

Gas law equations enable us to describe the behavior of gases under varying conditions of pressure, temperature, and volume. One of the most fundamental equations is the ideal gas law, given by $PV=nRT$. Here, $P$ is the pressure, $V$ is the volume, $n$ is the number of moles of gas, $R$ is the ideal gas constant, and $T$ is the temperature in Kelvin. In practice, when any of three variables $P$, $V$, or $T$ are known, along with the quantity $n$, we can rearrange this equation to solve for the unknown variable. For example, to find the volume of a gas, we would rearrange to $V=\frac{nRT}{P}$, and plug in the appropriate values for $n$, $R$, $T$, and $P$.

Calculating Gas Volume

Calculating the volume of a gas often stems from manipulation of the ideal gas law. Once you have the pressure, temperature, and moles of gas, you can calculate the volume using the rearranged ideal gas equation: $V=\frac{nRT}{P}$. By substituting in the values for the number of moles $n$, the ideal gas constant $R$, and the temperature in Kelvin $T$, the volume $V$ can be isolated and calculated. It’s important to make sure all the units are consistent, particularly that pressure is in Pascals (Pa), volume is in cubic meters (m³), and temperature is in Kelvin (K), to ensure the ideal gas constant units of $J/(molK)$ are compatible.

Pressure-Temperature Relationship

For a fixed amount of gas at a constant volume, its pressure and temperature are directly proportional, as described by Gay-Lussac's Law. This principle is embedded in the ideal gas law and is integral to solving problems where the temperature changes, leading to a change in pressure, or vice versa. When temperature increases, so does pressure, if the volume is kept constant. In the context of our problem, to find the amount of gas that has leaked from a tank due to temperature and pressure changes, we must first understand this relationship. It allows us to accurately calculate the change in moles of gas, taking into account that a decrease in temperature would result in a lower pressure, given by the equation $\frac{P_1}{T_1}=\frac{P_2}{T_2}$ where $P_1$ and $T_1$ are the initial pressure and temperature, and $P_2$ and $T_2$ are the final pressure and temperature respectively.