Question

A sample of nitrogen gas has a volume of \(1.00 \mathrm{~L}\) at STP. What is the temperature in \({ }^{\circ} \mathrm{C}\) if the volume is \(10.0 \mathrm{~L}\) at 2.00 atm?

Short answer

The final temperature is 5187°C.

Step-by-step solution

Understanding STP conditions

Standard temperature and pressure (STP) means a temperature of 0°C (273 K) and a pressure of 1 atm. So initially, the gas sample is at 1 atm and 273 K.

Applying the Combined Gas Law

The Combined Gas Law states \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). We need to find the final temperature \( T_2 \). We know \( P_1 = 1 \; \text{atm}, V_1 = 1.00 \; \text{L}, T_1 = 273 \; K \). The final conditions are \( P_2 = 2.00 \; \text{atm}, V_2 = 10.0 \; \text{L} \).

Plugging in the known values

Substitute the values into the Combined Gas Law: \[ \frac{1 \; \text{atm} \times 1.00 \; \text{L}}{273 \; \text{K}} = \frac{2.00 \; \text{atm} \times 10.0 \; \text{L}}{T_2} \].

Calculating \( T_2 \)

Rearrange the equation to solve for \( T_2 \): \[ T_2 = \frac{2.00 \; \text{atm} \times 10.0 \; \text{L} \times 273 \; \text{K}}{1 \; \text{atm} \times 1.00 \; \text{L}} \] Calculate: \[ T_2 = \frac{5460 \; \text{atm} \cdot \text{L} \cdot \text{K}}{1 \; \text{atm} \cdot \text{L}} = 5460 \; \text{K} \].

Converting Kelvin to Celsius

To convert the temperature from Kelvin to Celsius, use the formula \( T_{\circ \text{C}} = T_\text{K} - 273 \). Thus, \( T_{\circ \text{C}} = 5460 \; \text{K} - 273 \; \text{K} = 5187 \; ^{\circ} \text{C} \).

Key concepts

Gas Laws

Gas laws are essential principles in chemistry and physics that describe the behavior of gases. They explain how changes in conditions like pressure, volume, and temperature affect gases. One of the most comprehensive is the Combined Gas Law. This law combines three simpler laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law. By integrating these, the Combined Gas Law provides a relationship between pressure, volume, and temperature:

• Boyle's Law: States that pressure and volume are inversely proportional when temperature is constant.
• Charles's Law: Indicates volume and temperature are directly proportional with constant pressure.
• Gay-Lussac's Law: Shows pressure and temperature are directly proportional at constant volume.

The formula for the Combined Gas Law is:\[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]In this equation, \(P\) represents pressure, \(V\) volume, and \(T\) temperature in Kelvin. It's very useful for finding one variable when the others change.

STP Conditions

STP conditions stand for "Standard Temperature and Pressure," which are universal reference points for scientists when measuring gases. At STP, the temperature is 0°C, which is equivalent to 273.15 Kelvin, and the pressure is 1 atmosphere (atm). These conditions are very helpful in creating a benchmark for comparing the properties of gases and conducting experiments.
When a problem states that a gas is at STP, it provides a starting point to apply various gas laws. Understanding STP conditions allows you to plug in known values into equations and solve for unknowns. This consistency across experiments makes data comparable. So, if you're ever given that a gas is at STP, remember to use 273 K for temperature and 1 atm for pressure to simplify your calculations.

Temperature Conversion

Temperature conversion, especially between Celsius and Kelvin, is a crucial skill in chemistry. This is because gas laws require temperature to be in Kelvin for accurate calculations. To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature:\[T_{K} = T_{°C} + 273.15\]
When you need to convert back from Kelvin to Celsius, subtract 273.15:\[T_{°C} = T_{K} - 273.15\]This conversion is necessary because Kelvin is an absolute scale with zero as the absolute zero point, unlike Celsius which is based on the freezing and boiling points of water. Keeping this conversion in mind ensures you're using the correct scale in your gas law equations, avoiding common mistakes.

Pressure and Volume Relationships

The relationship between pressure and volume is a key aspect of gas behavior described by Boyle's Law. According to this law, if the temperature of a gas remains constant, the pressure and volume of the gas are inversely related. This means that if the volume of a gas increases, its pressure decreases, and vice versa, provided the temperature does not change.
This inverse relationship can be represented by the formula:\[P_1 V_1 = P_2 V_2\]As with many gas laws, the measurement of pressure in atm and volume in liters is common. Understanding how pressure and volume interact helps us predict how gases will respond when confined to different sizes of containers or when pressure is applied.

• Increasing volume leads to decreased pressure (if temperature is unchanged).
• Decreasing volume leads to increased pressure (if temperature is unchanged).

These principles are critical when manipulating gases in practical applications and when solving complex gas problems.