Question

Many gases can be modeled by the Ideal Gas Law, \(P V=n R T\), which relates the temperature \((T,\) measured in kelvins ( \(\mathrm{K}\) )), pressure ( \(P\), measured in pascals (Pa)), and volume ( \(V\), measured in \(\mathrm{m}^{3}\) ) of a gas. Assume the quantity of gas in question is \(n=1\) mole (mol). The gas constant has a value of \(R=8.3 \mathrm{m}^{3} \mathrm{Pa} / \mathrm{mol}-\mathrm{K}\) 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

Short answer

Answer: Part b (Isobars).

Step-by-step solution

Rearrange the Ideal Gas Law

Rearrange the Ideal Gas Law formula to find the expression for pressure (P) in terms of temperature (T) and volume (V): \(P=\frac{nRT}{V}\).

Choose some values for T

We need to plot some isotherms for temperatures between 0 à 900K. So, let's choose some values for T, for example: 300K, 600K, and 900K.

Calculate P(V) for each chosen T

Keeping the chosen values of T constant, calculate P as a function of V within the range \(0<V \leq 0.5\) and plot the level curves (isotherms). b. Isobars:

Rearrange the Ideal Gas Law

Rearrange the Ideal Gas Law formula to find the expression for temperature (T) in terms of pressure (P) and volume (V): \(T=\frac{PV}{nR}\).

Choose some values for P

We need to plot some isobars for pressure values between 0 à 100,000Pa. So, let's choose some values for P, for example: 25,000Pa, 50,000Pa, and 75,000Pa.

Calculate T(V) for each chosen P

Keeping the chosen values of P constant, calculate T as a function of V within the range \(0<V \leq 0.5\) and plot the level curves (isobars). c. Volume Level Curves:

Rearrange the Ideal Gas Law

Rearrange the Ideal Gas Law formula to find the expression for volume (V) in terms of temperature (T) and pressure (P): \(V=\frac{nRT}{P}\).

Choose some values for V

We need to plot level curves for volume values between 0 à 0.5 cubic meters. So, let's choose some values for V, for example: 0.1 m³, 0.3 m³, and 0.5 m³.

Calculate T(P) for each chosen V

Keeping the chosen values of V constant, calculate T as a function of P within the range \(0<P \leq 100,000\) and plot the level curves for the volume surface.

Key concepts

Isotherms

When studying the behavior of gases, isotherms provide a graphical way to understand how temperature remains constant within a specific setting. According to the Ideal Gas Law, which is stated as PV=nRT, isotherms are represented on a graph where temperature is held constant. In simple terms, isotherms are like contour lines on a map that show areas of equal temperature.

Isotherms are particularly useful when visualizing the relationship between pressure (P) and volume (V) of a gas when the gas is kept at a constant temperature. On a P-V graph, each isotherm will appear as a curve, and by selecting specific temperatures, such as 300K, 600K, and 900K, we effectively see how the pressure of the gas changes with volume at these temperatures. Isotherms are downward sloping on a P-V graph, indicating that as volume increases, the pressure decreases, and vice versa, adhering to Boyle's Law, one of the gas laws which states that pressure and volume are inversely proportional when temperature is held constant.

Isobars

Isobars, not to be confused with meteorological terms related to weather, in physics and particularly in thermodynamics, refer to curves that connect points of constant pressure in a given system. Sticking with the Ideal Gas Law, when we rearrange to solve for temperature as a function of pressure and volume, T = PV/nR, isobars are the plots we get by keeping the pressure fixed.

The practical value of observing isobars comes from their ability to showcase how temperature varies with volume at stable pressures. When graphing isobars on a T-V graph, one can see how at certain pressures, such as 25,000Pa, 50,000Pa, and 75,000Pa, the gas's temperature will rise as the volume increases. This is in line with Charles's Law, which states that the volume of a gas is directly proportional to its temperature when pressure is kept constant. Isobars, therefore, slope upwards on a T-V graph, indicating a positive correlation between volume and temperature under a set pressure.

Calculus Level Curves

In the realm of calculus, level curves are crucial tools used to represent multivariable functions visually. Level curves, which are also referred to as contour lines in the context of maps, depict points where a function has the same value – hence the term 'level'. In the case of the Ideal Gas Law, the level curves represent all the states (pressure, volume, and temperature) in which the gas behaves according to the same values of the dependent variable.

Whether we plot temperature, pressure, or volume as a function of the other two variables, these curves enable us to understand complex relationships between the variables without getting tangled in three-dimensional graphs. The concept of level curves extends beyond physics to many fields that involve spatial data, such as geography, economics, and optimization problems. Visualizing level curves allows students to understand the interplay between variables in a tangible way and enhances their ability to comprehend multidimensional data.