Question

When the pressure on \(1200 \mathrm{ml}\) of a gas is increased from \(70 \mathrm{~cm}\) to \(120 \mathrm{~cm}\) of mercury at constant temperature, the new volume of the gas will be (A) \(400 \mathrm{ml}\) (B) \(600 \mathrm{ml}\) (C) \(700 \mathrm{ml}\) (D) \(500 \mathrm{ml}\)

Short answer

Using Boyle's Law, \(P_1V_1 = P_2V_2\), where \(P_1 = 70 \: \mathrm{cm\: Hg}\), \(P_2 = 120 \: \mathrm{cm\: Hg}\), and \(V_1 = 1200 \: \mathrm{ml}\). Solve for the new volume, \(V_2\): \(V_2 = \frac{P_1V_1}{P_2} = \frac{70 \times 1200}{120} = 700 \: \mathrm{ml}\). The correct answer is (C) \(700 \mathrm{~ml}\).

Step-by-step solution

Remember Boyle's Law formula

Boyle's Law states that for a constant temperature and mass of gas, the pressure multiplied by the volume is constant. mathematically, Boyle's law is expressed as: \(P_1V_1 = P_2V_2\) Where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume.

Convert given pressure values to the same unit

The given pressure values are in cm of mercury. To make the calculations, we need to convert them to the same units (for example, Pascals or atm). However, in this problem, we don't need to worry about the units, because the pressure is given in the same unit (cm of mercury), and it will cancel out in the calculation.

Substitute the values into the formula

Let's first write down the initial and final pressure values, as well as the initial volume of the gas: \(P_1 = 70 \: \mathrm{cm\: Hg}\) \(P_2 = 120 \: \mathrm{cm\: Hg}\) \(V_1 = 1200 \: \mathrm{ml}\) Now, use the Boyle's Law formula: \(P_1V_1 = P_2V_2\) \(70 \: \mathrm{cm\: Hg} \times 1200 \: \mathrm{ml} = 120 \: \mathrm{cm\: Hg} \times V_2\)

Solve for the new volume (V2)

We need to find the new volume (V2), so we can now isolate V2 and solve for it: \(V_2 = \frac{P_1V_1}{P_2}\) Now, plug in the given values: \(V_2 = \frac{70 \: \mathrm{cm\: Hg} \times 1200 \: \mathrm{ml}}{120 \: \mathrm{cm\: Hg}}\)

Calculate the new volume and check the answer

Perform the arithmetic to find the new volume: \(V_2 = \frac{(70 \times 1200)}{120}\) \(V_2 = \frac{84000}{120}\) \(V_2 = 700 \: \mathrm{ml}\) So, the new volume of the gas is \(700 \: \mathrm{ml}\). Therefore, the correct answer is (C) \(700 \mathrm{~ml}\).

Key concepts

Pressure-Volume Relationship

Boyle's Law is key to understanding the pressure-volume relationship in gases. This law tells us how pressure and volume interact when we hold temperature constant, providing a clear mathematical framework for predicting how these properties change.

Essentially, Boyle's Law states that pressure and volume are inversely proportional. This means if one increases, the other decreases, provided temperature does not change. To spell this out:

• If you compress a gas (reduce its volume), its pressure goes up.
• If you expand a gas (increase its volume), its pressure goes down.

Understanding this relationship helps in explaining behaviors and phenomena in gases. The inverse relationship is captured by the formula: \[ P_1V_1 = P_2V_2 \],where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the pressure and volume after a change occurs. This equation shows that the product of pressure and volume stays constant, demonstrating the inverse relationship.

Gas Laws

Gas laws encompass several principles governing the behavior and properties of gases. These laws, including Boyle's Law, Charles's Law, and Avogadro's Law, form the foundation for understanding how gases respond to changes in pressure, volume, and temperature.

Boyle’s Law specifically addresses the behavior of gases when only pressure and volume are altered while temperature remains constant. Here are key points for gas laws:

• They allow predictions of gas behavior under different conditions.
• Each law isolates one variable (pressure, volume, or temperature) to see its effect, assuming other factors stay constant.
• Combined, these laws lead to the ideal gas law, a more comprehensive equation: \[ PV = nRT \].

Understanding these laws provides the framework to solve many practical problems involving gases, like adjusting tire pressure or describing the behavior of gases under various industrial conditions.

Constant Temperature

Constant temperature conditions, also known as isothermal conditions, are crucial when applying Boyle's Law. When temperature stays unchanged, it allows us to focus on how pressure and volume alone interact, without worrying about temperature introducing variability.

Here's why constant temperature is essential in this context:

• It eliminates the effect of thermal energy changes on gas molecules, which can influence pressure and volume.
• It simplifies the mathematical model, allowing pressure and volume to be the primary variables of interest.
• In real-world scenarios, constant temperature is assumed in controlled environments or slow processes where temperature change is minimal.

This simplicity provides a clear view of the inverse relationship between pressure and volume. For example, in experiments or calculations like determining a gas's new volume when compressed, maintaining constant temperature ensures valid application of Boyle's Law.