Question

Which of the following statements is(are) true? a. If the number of moles of a gas is doubled, the volume will double, assuming the pressure and temperature of the gas remain constant. b. If the temperature of a gas increases from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\), the volume of the gas would double, assuming that the pressure and the number of moles of gas remain constant. c. The device that measures atmospheric pressure is called a barometer. d. If the volume of a gas decreases by one half, then the pressure would double, assuming that the number of moles and the temperature of the gas remain constant.

Short answer

Statements a, c, and d are true, while statement b is false.

Step-by-step solution

Statement a: Doubling moles when pressure and temperature are constant

To verify this statement, you need to consider the Ideal Gas Law: \(PV = nRT\) where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. Since we are assuming that the pressure (P) and temperature (T) remain constant, and the number of moles (n) doubles, the equation looks like this: \(P(V_1) = n_1RT\) and \(P(V_2) = n_2RT\) Now we can compare the two equations: \(\frac{PV_2}{PV_1} = \frac{n_2RT}{n_1RT}\) As \(n_2 = 2n_1\), we get: \(\frac{V_2}{V_1} = \frac{2n_1}{n_1} = 2\) Thus, the volume (V) doubles when the number of moles doubles, assuming the pressure and temperature remain constant.

Statement b: Temperature increase and volume doubling

We again refer to the Ideal Gas Law as follows: \(P(V_1) = nR(T_1)\) and \(P(V_2) = nR(T_2)\) Since the pressure (P) and the number of moles (n) remain constant, we can compare the two equations by introducing the given temperature values: \(\frac{V_2}{V_1} = \frac{T_2}{T_1}\) Here, we need to convert the given temperatures to Kelvin: \(T_1 = 25^{\circ} \mathrm{C} + 273 = 298 \,\mathrm{K}\) \(T_2 = 50^{\circ} \mathrm{C} + 273 = 323 \,\mathrm{K}\) Now we can substitute the values: \(\frac{V_2}{V_1} = \frac{323}{298} \neq 2\) The statement is false as the volume does not double when the temperature increases from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\).

Statement c: Device measuring atmospheric pressure

This statement asks for the name of a device that measures atmospheric pressure. The device mentioned, a barometer, is a correct example of such a device. Therefore, statement c is true.

Statement d: Pressure increase and volume decrease

This statement can be verified using Boyle's Law, which is a simplified version of the Ideal Gas Law when the number of moles and temperature are constant: \(P_1V_1 = P_2V_2\) The given information states that the volume decreases by half. We can rewrite the equation as: \(\frac{P_2}{P_1} = \frac{V_1}{V_2}\) As \(V_2 = \frac{1}{2}V_1\), we have: \(\frac{P_2}{P_1} = \frac{V_1}{\frac{1}{2}V_1} = 2\) Thus, statement d is true, the pressure would double if the volume decreases by half, while the number of moles and temperature remain constant. In conclusion, statements a, c, and d are all true, while statement b is false.

Key concepts

Boyle's Law

Understanding Boyle's Law is crucial when studying the properties of gases. Boyle's Law describes how the pressure and volume of a gas are related when the temperature and amount of gas are fixed. Specifically, it states that the pressure of a gas is inversely proportional to its volume. This means that as the volume of a gas decreases, its pressure increases and vice versa. Mathematically, this relationship is expressed as:\[P_1V_1 = P_2V_2\]- \(P\) represents pressure.- \(V\) represents volume.- Subscripts \(1\) and \(2\) represent initial and final states.In practical terms, imagine squeezing a balloon. When you compress the balloon, its volume decreases, and you observe that the internal pressure rises. This relationship is the cornerstone of many practical applications, such as understanding how syringes work or predicting the behavior of gases under compression.

Barometer

The barometer is an essential device for measuring atmospheric pressure. This device helps us understand changes in weather conditions and is crucial for fields like meteorology. A mercury barometer, for instance, consists of a glass tube filled with mercury and closed at one end, placed upside down in a mercury-filled reservoir. As atmospheric pressure varies, it presses on the reservoir, causing the mercury level in the tube to rise or fall. Here are some key points about barometers:

• They are used to predict weather changes.
• High atmospheric pressure often correlates with clear weather, while low pressure can indicate storms.
• They provide valuable information for air travel and other industries.

Whether electronic or traditional, barometers remain vital tools for meteorologists and aviation professionals alike.

Volume-Pressure Relationship

The relationship between volume and pressure is fundamental to understanding the behavior of gases, especially under changing conditions. This relationship is highlighted in gas laws, particularly Boyle's Law, as we discussed earlier. When external pressure is applied to a gas, its volume changes accordingly, assuming temperature and the number of gas molecules remain constant. Some key aspects to remember about this relationship are:

• An increase in pressure will cause a decrease in volume.
• This behavior is reversible: reducing pressure allows the volume to expand back.
• The relationship is pivotal in many natural and industrial processes.

Grasping the volume-pressure relationship helps predict and manipulate the behavior of gases in various scenarios, from natural meteorological phenomena to engineering systems that rely on gas compression.