Question

\(750 \mathrm{ml}\) of gas at 300 torr pressure and \(50^{\circ} \mathrm{C}\) is heated until the volume of gas is \(2000 \mathrm{ml}\) at a pressure of 700 torr. What is the final temperature of the gas?

Short answer

The final temperature of the gas is approximately \(601.62\) K.

Step-by-step solution

Converting the given values

First step is to convert all given values to proper units. The pressure should be in atm (atmospheres), the volume in liters (L), and the temperature in Kelvin (K). Initially: - \( P_1 = 300\) torr (convert this to atm by dividing by 760 to get \(0.3947\) atm). - \( V_1 = 750\) ml (convert this to L by dividing by 1000 to get \(0.75\) L). - \( T_1 = 50\) degrees C (convert this to K by adding 273.15 to get \(323.15\) K). Finally: - \( P_2 = 700\) torr (convert this to atm by dividing by 760 to get \(0.9211\) atm). - \( V_2 = 2000\) ml (convert this to L by dividing by 1000 to get 2 L). - \( T_2 \) is what we need to find.

Rearranging the Combined Gas Law equation

We rearrange the Combined Gas Law to solve for \( T_2 \): \( T_2 = \frac{P_2 \cdot V_2 \cdot T_1}{P_1 \cdot V_1} \).

Substituting the known values into the equation

We substitute the known values into the rearranged equation to solve for \( T_2 \): \( T_2 = \frac{(0.9211 \text{ atm} \cdot 2 \text{ L} \cdot 323.15 \text{ K})}{(0.3947 \text{ atm} \cdot 0.75 \text{ L})} \).

Calculating the final temperature

We do the math to find \( T_2 \), ensuring that the units of atm and L cancel out, leaving \( T_2 \) in Kelvin. \( T_2 \approx 601.62 \) K So, the final temperature of the gas is approximately \(601.62\) K.

Key concepts

Gas Laws

The behavior of gases is described by different laws that define the relationships between pressure, volume, and temperature. The Combined Gas Law is particularly useful when dealing with situations where both pressure and volume change while temperature shifts as well. This law consolidates Boyle's, Charles's, and Gay-Lussac's laws and can be expressed as:\[ \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2} \]This equation shows that the ratio of the product of pressure and volume to temperature remains constant in a closed system. When using this law, it is crucial to ensure that all units are consistent. This means keeping pressure in atmospheres (atm), volume in liters (L), and temperature in Kelvin (K). By understanding the Combined Gas Law, students can predict how a gas will behave when subjected to changes in temperature, pressure, and volume in real-world situations.

Temperature Conversion

When working with gas laws, converting temperatures to Kelvin is essential. Kelvin is the absolute temperature scale needed for these calculations because it starts at absolute zero, making it directly proportional to the kinetic energy of the particles. This is unlike Celsius or Fahrenheit.To convert a temperature from Celsius to Kelvin, simply add 273.15:- From Celsius to Kelvin: \( T(K) = T(°C) + 273.15 \)This conversion ensures all parts of the gas law equations are compatible and accurate. It also reinforces the connection between temperature changes and energy changes within the gas.

Unit Conversion

Unit conversion is a fundamental step when solving gas law problems. Different measurements, like pressure, volume, and temperature, must be expressed in consistent units to apply the formulas correctly. Here's how you do it:

• **Volume Conversion**: Milliliters (mL) to Liters (L): Divide by 1000. For example, \(750 \text{ mL} = 0.75 \text{ L}\).
• **Pressure Conversion**: Torr to Atmospheres (atm): Divide by 760. For example, \(300 \text{ torr} = 0.3947 \text{ atm}\).
• **Temperature Conversion**: From Celsius to Kelvin: Add 273.15. For example, \(50^{\circ} \text{C} = 323.15 \text{ K}\).

Converting units accurately is critical to ensuring calculations are correct, which further aids in preventing errors in more complex equations.

Pressure Units

Pressure is a measure of force applied per unit area. Several units are used to measure gas pressure, but the most common are atmospheres (atm) and torr. In scientific equations like the Combined Gas Law, pressure must often be in atmospheres to maintain dimensional consistency.- **Torr to Atmosphere Conversion**: Dividing the pressure in torr by 760 converts it to atmospheres.Example:\( 700 \text{ torr} \div 760 = 0.9211 \text{ atm} \).This conversion is part of the broader process of ensuring that all units are consistent, which is necessary for the calculations to be valid. Pressure conversions are an integral step for accurate problem-solving in gas laws.

Volume Units

Volume is a measure of the space a gas occupies, and for gas laws, volume is usually measured in liters. It's important to convert volumes expressed in milliliters to liters, as any inconsistency can lead to errors in calculations. Here's how you do it:- **Milliliters to Liters Conversion**: Divide the volume in milliliters by 1000.Example:\( 2000 \text{ mL} = 2 \text{ L} \).Using liters ensures that the volume unit is compatible with other variables in gas law equations. Consistent volume units contribute to accurate application and interpretation of the laws that govern gas behavior.