Question

Rewrite the ideal gas law solving for \(n\). Also show how all units cancel to leave you with just units of moles.

Short answer

To solve the ideal gas law for the number of moles (n), rearrange the equation: \(PV = nRT\), as \(n = \frac{PV}{RT}\). When considering the units of pressure (P), volume (V), gas constant (R), and temperature (T), we see that they cancel out to leave only the unit of moles (mol) as follows: \(\frac{Pa \cdot m^3}{(\frac{Pa \cdot m^3}{mol\cdot K}) \cdot K} = mol\).

Step-by-step solution

Ideal gas law equation

The ideal gas law equation is given as: \(PV = nRT\) Step 2: Rearrange the equation to solve for n

Rearrange the ideal gas law equation for n

To solve for n, we need to isolate n on one side of the equation. Divide both sides of the equation by RT: \[n = \frac{PV}{RT}\] Step 3: Show cancellation of units

Unit cancellation

Now, let's consider the units in the equation while ensuring they cancel out to give us just units of moles. The units of pressure (P) can be pascal (Pa) or atmospheres (atm), volume (V) is usually in liters (L) or cubic meters (m³), the gas constant (R) can have different values depending on the units: 8.314 J/(mol·K) or 0.08206 L·atm/(mol·K) and temperature (T) is in Kelvin (K). Expressing the ideal gas law in SI units: \[\frac{Pa \cdot m^3}{(\frac{J}{mol\cdot K}) \cdot K} = mol\] We need to show that J(Pa·m³) cancels out to leave the moles unit. 1 J (joule) is the standard unit of energy and can be defined as: \[1\,J = 1\,Pa \cdot m^3\] With this in mind, the units in the equation become: \[\frac{Pa \cdot m^3}{(\frac{Pa \cdot m^3}{mol\cdot K}) \cdot K} = mol\] As we can see, the units of Pa and m³ in both the numerator and denominator cancel each other out, leaving us with the unit of moles: \[\frac{1}{\frac{1}{mol}} = mol\] Therefore, we have successfully shown that the ideal gas law solved for n and all units cancel out to leave us with just units of moles.

Key concepts

Moles in Chemistry

To understand the Ideal Gas Law and its application, it is important to understand the concept of moles. In chemistry, a mole is a unit that measures the amount of substance. A single mole contains exactly Avogadro's number of entities, which means approximately \(6.022 \times 10^{23}\) atoms, molecules, or ions. This number helps chemists convert between the mass of a substance and the number of particles it contains. Moles make it convenient to express chemical quantities and reactions, providing a bridge between the atomic scale and real-world measurements. When using the Ideal Gas Law equation \(PV = nRT\), moles serve as the crucial variable \(n\) representing the amount of gas present.

Unit Cancellation and Why It Matters

Unit cancellation is a critical concept when using equations like the Ideal Gas Law. It ensures that all units apart from those we are solving for cancel out, leading us to a correct and consistent result.Let's consider applying unit cancellation to the Ideal Gas Law, solving for moles \(n\). Given the equation \(n = \frac{PV}{RT}\), each symbol involves specific units:- \(P\), pressure, might be in pascal (Pa) or atmospheres (atm).- \(V\), volume, is frequently in liters (L) or cubic meters (m³).- \(R\), the gas constant, changes depending on units: 8.314 J/(mol·K) or 0.08206 L·atm/(mol·K).- \(T\), temperature, is in Kelvin (K).In SI units, we simplify: \(\frac{Pa \cdot m^3}{(\frac{J}{mol \cdot K}) \cdot K} = mol\). The relationship \(1\,J = 1\,Pa \cdot m^3\) allows the units in the formula to cancel smoothly, leaving only the desired unit of moles. Proper unit cancellation guarantees that our mathematical manipulation doesn’t affect the physical meaning of the equation.

Rearranging Equations: Solving for Variables

Rearranging equations is a vital skill in scientific calculations, as it lets us solve for unknown variables. When tackling the Ideal Gas Law \(PV = nRT\), rearranging to find \(n\) involves isolating \(n\) on one side of the equation.To achieve this, you can divide both sides by \(RT\), resulting in:\[n = \frac{PV}{RT}\]This manipulation shifts the constants \(R\) and \(T\) away from \(n\), isolating it to show how it's affected by pressure \(P\) and volume \(V\). By practicing equation rearrangement, you build flexibility in problem-solving across different scientific areas. It empowers you to adapt known equations to the specific data or unknowns you might encounter, being a cornerstone of scientific method and inquiry.