Question

Given that \(P V=n R T\), solve for the unknown quantity. $$ \begin{array}{|l|l|l|l|l|} \hline \multicolumn{1}{|c|}{\boldsymbol{P}} & \multicolumn{1}{c|}{\boldsymbol{V}} & \multicolumn{1}{c|}{\boldsymbol{n}} & \multicolumn{1}{c|}{\boldsymbol{R}} & \multicolumn{1}{c|}{\boldsymbol{T}} \\ \hline 3.55 & 1.75 & 0.205 & 0.08206 & ? \\ \hline 1.00 & 22.5 & ? & 0.08206 & 298 \\ \hline 0.125 & ? & 3.00 & 0.08206 & 535 \\ \hline ? & 6.25 & 1.57 & 0.08206 & 343 \\ \hline \end{array} $$

Short answer

After calculation, the solutions for the unknowns are approximately \(T=102.08\) K for scenario 1, \(n=9.11\) mol for scenario 2, \(V=10421.92\) L for scenario 3 and \(P=6.74\) atm for scenario 4.

Step-by-step solution

Solve for the first unknown T

For the first scenario, the unknown factor is \(T\), the temperature. Re-arranging the ideal gas equation to find \(T\) gives the equation as \(T=(PV)/(nR)\). Here, \(P=3.55\) atm, \(V=1.75\) L, \(n=0.205\) mol, and \(R=0.08206\) L.atm/K.mol. Substituting these values gives \(T=(3.55*1.75)/(0.205*0.08206)\).

Solve for the second unknown n

For the second scenario, the unknown factor is \(n\), the number of moles. Re-arranging the ideal gas equation to find \(n\) gives the equation as \(n=(PV)/(RT)\). Here, \(P=1.00\) atm, \(V=22.5\) L, \(R=0.08206\) L.atm/K.mol and \(T=298\) K. Substituting these values gives \(n=(1.00*22.5)/(0.08206*298)\).

Solve for the third unknown V

For the third scenario, the unknown factor is \(V\), the volume. Re-arranging the ideal gas equation to find \(V\) gives the equation as \(V=(nRT)/(P)\). Here, \(P=0.125\) atm, \(n=3.00\) mol, \(R=0.08206\) L.atm/K.mol and \(T=535\) K. Substituting these values gives \(V=(3.00*0.08206*535)/(0.125)\).

Solve for the fourth unknown P

For the fourth scenario, the unknown factor is \(P\), the pressure. The ideal gas equation is already arranged for \(P\), which is \(P=(nRT)/(V)\). Here, \(V=6.25\) L, \(n=1.57\) mol, \(R=0.08206\) L.atm/K.mol and \(T=343\) K. Substituting these values gives \(P=(1.57*0.08206*343)/(6.25)\).

Key concepts

Solving for Temperature in Ideal Gas Law

Understanding how to find the temperature of a gas using the ideal gas law is crucial in the field of chemistry and physics. The ideal gas law equation is given by \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for moles of gas, \(R\) for the ideal gas constant, and \(T\) for temperature in Kelvin.

When solving for temperature, the formula is manipulated to \(T = \frac{PV}{nR}\). In practice, you must ensure that your units are consistent. For example, pressure should be in atmospheres (atm) and volume in liters (L) when using the common value of \(R = 0.08206 \frac{L \cdot atm}{K \cdot mol}\).

To solve a problem, substitute the known values into the rearranged equation. It's a straightforward calculation, but double-check your units and conversion to Kelvins if needed. Remember, temperature values in the ideal gas law should always be in Kelvin to maintain consistency and accuracy.

Calculating Moles Using Ideal Gas Equation

The ideal gas law is also a valuable tool for calculating the amount of substance present in a given gas sample. When calculating moles, the formula re-arranges to \(n = \frac{PV}{RT}\).

This re-arrangement provides a direct approach to determine the number of moles \(n\) if the conditions of pressure \(P\), volume \(V\), and temperature \(T\) are known. The same precautions about units apply here; ensure that you are using the appropriate units for pressure and volume and that the temperature is in Kelvin.

Sample Calculation:

For an example using the given conditions in the exercise, you would carry out the calculation using those specific values. This context directly applies when students need to find the quantity of a gas involved in a reaction or when determining the yield from a given reaction.

Determining Volume from Ideal Gas Law

In scenarios where the volume of gas needs to be found, the ideal gas law informs us well. If we rearrange the equation to solve for \(V\), it becomes \(V = \frac{nRT}{P}\).

This manipulation allows us to find the volume occupied by a certain number of moles of a gas at a known temperature and pressure. As always, paying attention to units is crucial for accurate calculations, since using the value of \(R\) in liter-atmospheres per mole-Kelvin requires that the other units are compatible.

To determine the volume, substitute the required values into the equation. This is particularly important in laboratory settings or industrial applications where the volume of gas tanks or reactors needs to be calculated.

Pressure Calculations with Ideal Gas Law

Calculating the pressure of a gas system is another application of the ideal gas law. This time, the equation for pressure will be \(P = \frac{nRT}{V}\).

When you want to find the pressure exerted by a gas in a container with a known volume, number of moles, and temperature, this form of the equation is used. Correct unit usage remains essential here: pressure in atmospheres, volume in liters, and temperature in Kelvin pair well with the \(R\) constant value used in earlier examples.

This aspect of the ideal gas law is particularly relevant in fields like environmental science and engineering, where understanding the pressure of gases within a system is essential for designing safety mechanisms and ensuring the integrity of pressure vessels.