Question

Describe the curve one would obtain by plotting pressure versus volume for an ideal gas in which the temperature and number of moles of gas are held constant.

Short answer

The curve obtained by plotting pressure versus volume for an ideal gas with constant temperature and number of moles is a hyperbolic isotherm, representing the equation \(PV = k\). This curve shows an inverse relationship between pressure and volume and is located in the first quadrant of a P-V diagram. As pressure decreases, volume increases, and the curve approaches both the x and y-axes asymptotically without actually touching them.

Step-by-step solution

Recall the ideal gas law equation

The ideal gas law equation is expressed as \(PV=nRT\), where P stands for pressure, V stands for volume, n is the number of moles of gas, R is the universal gas constant, and T is temperature.

Identify the constant variables

In this problem, temperature (T) and the number of moles of gas (n) are constant. The universal gas constant (R) is also constant.

Rearrange the ideal gas law equation to express the relation between pressure (P) and volume (V)

Since the number of moles, temperature, and gas constant are constant, we can write the equation as \(PV = k\), where k is a constant representing the product of n, R, and T.

Describe the curve between pressure (P) and volume (V)

The equation \(PV = k\) represents a hyperbolic relationship between pressure and volume. This curve is an isotherm and shows how pressure decreases as volume increases (and vice versa) while maintaining a constant temperature. The curve will approach both the x and y-axes asymptotically but never actually touch them. In a P-V diagram, the curve will have a decreasing slope and be located in the first quadrant since both pressure and volume values are positive.

Key concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the behavior of ideal gases. It is depicted by the equation, \(PV = nRT\), where \(P\) represents the pressure exerted by the gas, \(V\) is the volume it occupies, \(n\) is the amount of substance in moles, \(R\) is the ideal gas constant, and \(T\) signifies the absolute temperature.

For a gas that behaves ideally, the product of its pressure and volume is directly proportional to the amount of gas (as measured in moles) and the absolute temperature. The ideal gas constant, \(R\), acts as a bridge connecting all these variables. It's important to note that when we say 'ideal,' we refer to a simplified model where the gas particles do not interact with each other and occupy no volume. However, this law provides a useful approximation for the behavior of real gases under many conditions.

When applying the Ideal Gas Law, if any of the variables except one are held constant, the remaining one can be varied to understand different physical scenarios of gas behavior. This forms the foundation of other related gas laws, including Boyle's Law which is explored further with the pressure-volume relationship.

Isotherms and Their Significance

An isotherm is a line on a graph that represents constant temperature conditions for a given process. In the context of the Ideal Gas Law, an isotherm can be visualized on a pressure-volume (P-V) graph that shows how pressure and volume of an ideal gas change relative to one another when the temperature is held constant.

The significance of an isotherm lies in its illustration of the interdependent relationship between pressure and volume under isothermal conditions—'iso' meaning equal and 'therm' referring to temperature. On a P-V diagram, each isotherm represents a different temperature, and the behavior of gases at these temperatures can be analyzed separately.

For students to better grasp this concept, imagine a scenario where the ideal gas within a container is maintained at a constant temperature. As the gas is compressed (volume decreases), its particles become more concentrated and collide against the walls more frequently and forcefully, thus increasing the pressure. Conversely, when the gas expands, the pressure drops. The graphical depiction of this behavior on an isotherm allows students to visualize the nature of these changes and understand that the product (\(PV\)) remains constant.

Pressure-Volume Relationship in Gases

The pressure-volume relationship, commonly known as Boyle's Law when temperature is constant, describes how the pressure of a given amount of ideal gas tends to increase as the volume decreases, and vice versa. Formally stated, \(P_1V_1 = P_2V_2\), where the subscripts 1 and 2 refer to the initial and final states of the gas.

Boyle's Law is a particular case of the Ideal Gas Law where temperature and the amount of gas are held constant. The relationship between pressure and volume is inversely proportional, meaning if one increases, the other must decrease proportionally to maintain a constant product. This inverse relationship results in a hyperbolic graph when you plot pressure against volume.

Examples in Everyday Life

• When you squeeze a closed, air-filled syringe, you reduce its volume, and the pressure inside increases, demonstrating Boyle's Law.
• A balloon inflating as it rises in the atmosphere is another example, where the external pressure decreases with altitude, allowing the balloon to expand in volume.

Understanding this fundamental concept allows students to predict how gases will behave when they are compressed or expanded, which is crucial in various applications ranging from breathing mechanisms to the operation of internal combustion engines.