Question

Ideal Gas Law Many gases can be modeled by the Ideal Gas Law, \(P V=n R T,\) which relates the temperature \((T,\) measured in Kelvin (K)), pressure ( \(P\), measured in Pascals (Pa)), and volume ( \(V\), measured in \(\mathrm{m}^{3}\) ) of a gas. Assume that the quantity of gas in question is \(n=1\) mole (mol). The gas constant has a value of \(R=8.3 \mathrm{m}^{3} \cdot \mathrm{Pa} / \mathrm{mol} \cdot \mathrm{K}.\) a. Consider \(T\) to be the dependent variable and plot several level curves (called isotherms) of the temperature surface in the region \(0 \leq P \leq 100,000\) and \(0 \leq V \leq 0.5.\) b. Consider \(P\) to be the dependent variable and plot several level curves (called isobars) of the pressure surface in the region \(0 \leq T \leq 900\) and \(0< V \leq 0.5.\) c. Consider \(V\) to be the dependent variable and plot several level curves of the volume surface in the region \(0 \leq T \leq 900\) and \(0 < P \leq 100,000.\)

Short answer

b. Isobars are curves with constant pressure in the \(TV\) plane. What are the formula and selected pressure levels for isobars? c. Isochors are curves with constant volume in the \(TP\) plane. What are the formula and selected volume levels for isochoric curves?

Step-by-step solution

Rearrange ideal gas law to solve for T

First, we should rearrange the ideal gas law to isolate the temperature variable. We have: \(T = \frac{PV}{nR}\). Since \(n = 1\) mole and \(R = 8.3 \frac{m^3 \cdot Pa}{mol \cdot K}\), we can simplify the expression to: \(T = \frac{PV}{8.3}\).

Define the plot's domain and desired isotherms

We are asked to plot isotherms in the region \(0 \leq P \leq 100,000\) and \(0 \leq V \leq 0.5\) with several temperature levels. Let's choose equally spaced temperature levels, like \(T= \{100, 200, 300, 400, 500, 600, 700\} K\).

Plot the isotherms

Now, we can plot the isotherms \(T\) in the \((P, V)\) space. For each temperature value \(T\), we can compute the contour points for the given \((P, V)\) values in the specified region using the simplified equation, \(T = \frac{PV}{8.3}\). #b. Isobars (Pressure as dependent variable)#

Rearrange ideal gas law to solve for P

We should rearrange the ideal gas law to isolate the pressure variable. We have: \(P = \frac{nRT}{V}\). Since \(n = 1\) mole and \(R = 8.3 \frac{m^3 \cdot Pa}{mol \cdot K}\), we can simplify the expression to: \(P = \frac{8.3T}{V}\).

Define the plot's domain and desired isobars

We are asked to plot isobars in the region \(0 \leq T \leq 900\) and \(0< V \leq 0.5\) with several pressure levels. Let's choose equally spaced pressure levels, like \(P= \{20,000, 40,000, 60,000, 80,000, 100,000\} Pa\).

Plot the isobars

Now, we can plot the isobars \(P\) in the \((T, V)\) space. For each pressure value \(P\), we can compute the contour points for the given \((T, V)\) values in the specified region using the simplified equation, \(P = \frac{8.3T}{V}\). #c. Isochoric curves (Volume as dependent variable)#

Rearrange ideal gas law to solve for V

We rearrange the ideal gas law to isolate the volume variable. We have: \(V = \frac{nRT}{P}\). Since \(n = 1\) mole and \(R = 8.3 \frac{m^3 \cdot Pa}{mol \cdot K}\), we can simplify the expression to: \(V = \frac{8.3T}{P}\).

Define the plot's domain and desired isochoric curves

We are asked to plot isochoric (constant volume) curves in the region \(0 \leq T \leq 900\) and \(0 < P \leq 100,000\) with several volume values. Let's choose equally spaced volume values, like \(V=\{0.1, 0.2, 0.3, 0.4, 0.5\} m^3\).

Plot the isochoric curves

Now, we can plot the isochoric curves \(V\) in the \((T, P)\) space. For each volume value \(V\), we can compute the contour points for the given \((T, P)\) values in the specified region using the simplified equation, \(V = \frac{8.3T}{P}\).

Key concepts

Temperature

Temperature is a critical factor in the Ideal Gas Law. It is represented by the letter \(T\), and is measured in Kelvin (K). The relationship between temperature and other variables like pressure and volume is essential to understanding gas behavior.

According to the Ideal Gas Law, as temperature increases, for a constant amount of gas and volume, pressure also increases. This occurs because the kinetic energy of molecules rises with temperature, causing them to exert more force.

It's important to always convert temperatures to Kelvin when using gas laws. Celsius and Fahrenheit generally don't apply here because Kelvin ensures absolute temperature measurements starting from absolute zero.

Pressure

Pressure, denoted as \(P\), is another fundamental concept in gas laws. It is measured in Pascals (Pa). Pressure is defined as the force applied per unit area by the gas particles colliding with the walls of their container. The Ideal Gas Law tells us that for a given amount of gas, if we increase the pressure, the temperature or the volume must respond in kind.

If you keep the volume constant and increase pressure, temperature must rise to maintain the balance set by the Ideal Gas Law \(PV = nRT\). This underlines the close relation between pressure, temperature, and volume.

Volume

Volume, represented by \(V\), refers to the space a gas occupies, measured in cubic meters (m³) in the context of the Ideal Gas Law. When considering gases, it's important to understand that they expand to fill their containers, and thus the volume is often dictated by the constraints of the environment.

The Ideal Gas Law illustrates that volume shares a direct relationship with temperature; if the pressure holds steady, an increase in temperature should result in a proportional increase in volume. This is why a balloon inflates when heated. Remember that volume is essential in calculating other gas properties using the formula \(PV = nRT\).

Isotherms

Isotherms are curves on a graph of pressure vs. volume (P-V graph) which represent levels of constant temperature. In the context of the Ideal Gas Law, an isotherm shows how pressure and volume interact when the temperature remains unchanged.

Understanding isotherms helps in visualizing how gases behave under constant temperature. It’s like observing different scenarios where a gas maintains the same kinetic energy but experiences varied pressures and volumes. In real-world scenarios, this helps predict how gases will react in controlled temperature environments.

Isobars

Isobars are lines on a graph indicating constant pressure. When creating these plots in scenarios like our exercise, one often uses the temperature and volume axes. These show how gases behave when pressure is kept constant while temperature and volume change.

Using isobars can help in understanding the expansion and contraction of gases. For instance, as temperature increases under constant pressure, the volume must also increase. This behavior helps explain why hot air balloons rise; the heated air inside causes the internal pressure to decrease initially, allowing the balloon to expand and float.

Isochoric curves

Isochoric curves, also known as isometric processes or constant-volume processes, are graphs where the volume remains fixed. They explore the relationship between pressure and temperature, helping understand gas behavior in a rigid, unchangeable volume.

Inside the Ideal Gas framework, the pressure increases as the temperature increases while maintaining constant volume. This behavior is crucial in understanding compression scenarios where volume cannot decrease, like in a sealed container subjected to heat. If you think about a can of aerosol paint, as it heats up, pressure increases because the gas inside cannot expand to occupy more space.