Problem 86 - College Algebra and Calculus: An Applied Approach

Question

Solve for the indicated variable. Ideal Gas Law Solve for \(T\) in \(P V=n R T\).

Step-by-Step Solution

Answer

The equation solved for \(T\) is \(T = \frac{PV}{nR}\).

Step 1: Identify the Variable to Solve For

From the equation, we can see that we need to solve for the temperature, \(T\).

Step 2: Isolate \(T\)

To get \(T\), we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by \(nR\). Doing this, we get: \(T = \frac{PV}{nR}\).

Step 3: Final Simplified Equation

After performing the operation in the previous step, we obtain the final equation: \(T = \frac{PV}{nR}\), which gives temperature \(T\) in terms of pressure \(P\), volume \(V\), number of moles \(n\), and gas constant \(R\).

Key Concepts

Temperature

Temperature is a central concept in the Ideal Gas Law as it relates to the kinetic energy of the gas particles. In the equation \(PV = nRT\), \(T\) represents the temperature of the gas, which must be measured in Kelvin to maintain consistency with the gas constant \(R\). The Kelvin scale is preferred because it starts at absolute zero, the theoretical point where particles have minimum thermal motion.

Understanding temperature in this context is essential because it directly affects the pressure and volume of a gas. As temperature increases, gas particles move more vigorously, leading to increased pressure if the volume is constant or increased volume if the pressure is constant.

• Temperature in Kelvin = Temperature in Celsius + 273.15
• Always ensure your temperature data is converted to Kelvin when using the Ideal Gas Law for calculations.

Simplifying the calculation into Kelvin ensures the relationship between pressure, volume, and the number of moles remains consistent.

Pressure

Pressure in the Ideal Gas Law equation \(PV = nRT\) is represented by \(P\). Pressure is the force exerted by gas particles colliding with the walls of their container. It is an indication of how often and how forcefully the particles hit the container walls.

Pressure is often measured in units such as atmospheres (atm), Pascals (Pa), or millimeters of mercury (mmHg). In calculations, it's important to convert pressure to the same unit as required by the gas constant \(R\) to ensure accuracy.

• Common Unit Conversions:
  • 1 atm = 101325 Pa
  • 1 atm = 760 mmHg
• Ensure unit consistency with \(R\) for correct results.

Pressure is closely linked with temperature and volume; an increase in temperature, for instance, causes particles to hit the walls more frequently, potentially increasing pressure if the volume isn't adjusted.

Volume

Volume, denoted as \(V\) in the Ideal Gas Law equation \(PV = nRT\), relates to the space occupied by a gas. The concept of volume is integral because it affects how gas particles move and interact.

Volume is usually measured in liters (L) or cubic meters (m³). Similar to pressure and temperature, ensuring that volume is measured in the consistent unit for the equation to work correctly is important. Most often, volume is measured in liters for applications involving the Ideal Gas Law.

• 1 cubic meter = 1000 liters
• Select your units based on the gas constant \(R\) to maintain uniformity.

Changes in volume can affect pressure and temperature, impacting how and where gas particles can move, which influences the overall energy dynamics within the system.

Number of Moles

The number of moles, represented by \(n\) in the Ideal Gas Law equation \(PV = nRT\), is a measure of the quantity of gas particles. It's an essential aspect since it directly relates to the mass and the number of molecules within the gas sample.

The concept of moles allows scientists to count particles in a tangible way, using Avogadro's number, which states that one mole is equal to approximately 6.022 x 10²³ particles. This concept is crucial for predicting how a gas will behave under different conditions of pressure, volume, and temperature.

• Avogadro's Principle relates the number of moles to the volume at standard temperature and pressure.
• Molar Mass can help relate the mass of the gas to the number of moles using the formula: Moles = Mass / Molar Mass.

The number of moles, along with the Ideal Gas Law, enables the calculation of unknown variables when some conditions of a gas system are specified.