Question

\(100 \mathrm{ml}\). of gas are enclosed in a cylinder under a pressure of 760 Torr. What would the volume of the same gas be at a pressure of 1520 Torr?

Short answer

The new volume of the gas at a pressure of 1520 Torr is 50 mL.

Step-by-step solution

Write down the given information

The initial volume, V1, is 100 mL. The initial pressure, P1, is 760 Torr, and the final pressure, P2, is 1520 Torr. We need to find the final volume, V2.

Write down Boyle's Law formula

Boyle's Law can be written as: \[P1V1 = P2V2\]

Substitute in the known values

We plug in the given information into the equation: \[760 \cdot 100 = 1520 \cdot V2\]

Solve for the final volume (V2)

Now we can solve for V2 by dividing both sides of the equation by 1520: \[V2 = \frac{760 \cdot 100}{1520} \]

Calculate the new volume

Calculate V2: \[V2 = \frac{76000}{1520} = 50\] So the new volume of the gas at a pressure of 1520 Torr is 50 mL.

Key concepts

Gas Laws

Gas laws are fundamental principles in chemistry and physics, explaining how gases behave under various conditions. They connect variables such as pressure, volume, and temperature. A prime example of these principles is Boyle's Law, Charles's Law, and the Ideal Gas Law. All these laws help predict how a gas will respond when changes occur, though they each focus on different aspects and variables.

• Boyle's Law: Relates pressure and volume while keeping temperature constant.
• Charles's Law: Connects volume and temperature, maintaining constant pressure.
• Ideal Gas Law: Combines pressure, volume, temperature, and the number of moles of a gas for a comprehensive understanding.

Understanding these laws is crucial as they assist in various applications, such as inflating balloons, using gas in engines, and even comprehending human respiration. Boyle's Law is particularly handy in our exercise, explaining the relationship between pressure and volume of a contained gas sample, assuming temperature remains constant.

Pressure and Volume Relationship

The pressure and volume relationship of a gas is a fascinating concept grounded in Boyle's Law. This relationship is inversely proportional, meaning if one increases, the other decreases, provided the temperature remains constant. This is vital in studying real-life scenarios where gas conditions change but temperature remains unchanged.

In mathematical terms, Boyle's Law is expressed as:\[ P_1V_1 = P_2V_2 \]This formula shows that the product of initial pressure and volume (\( P_1 \) and \( V_1 \)) will equal the product of final pressure and volume (\( P_2 \) and \( V_2 \)).

• When pressure increases, volume decreases.
• When pressure decreases, volume increases.

This relationship is prominent in the exercise where the gas volume is reduced from 100 mL to 50 mL as pressure doubles, reflecting the inverse nature of the relationship.

Ideal Gas Behaviors

The ideal gas behaviors are simplified models that predict how an 'ideal' gas would behave under certain conditions. However, real gases often exhibit slight deviations from these models, especially under high pressure or low temperature. Yet, under typical conditions, these models serve as a practical guide for understanding gas properties.

An ideal gas follows specific laws like the Ideal Gas Law, defined as:\[ PV = nRT \]where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature in Kelvin.

• Assumes no intermolecular forces between gas particles.
• Predicts consistent behavior in volume changes across different pressures.
• Often used in understanding basic principles before considering real-life complexities.

Understanding ideal gas behaviors helps in grasping core concepts and preparing for more advanced study of gas mechanics. Though our exercise applies Boyle’s Law specifically, acknowledging these ideal assumptions helps enhance comprehension of gas interactions.