Question

A sample of ammonia gas has a volume of \(50.0 \mathrm{~mL}\) at STP. What is the Celsius temperature if the volume is \(350.0 \mathrm{~mL}\) at \(350 \mathrm{~mm}\) Hg?

Short answer

The Celsius temperature is approximately 606.3°C.

Step-by-step solution

Identify Known Values

First, identify the given values and the conditions at which we need to determine the temperature. We have:- Initial volume, \( V_1 = 50.0 \, \text{mL} \)- Initial pressure at STP, \( P_1 = 760 \, \text{mm Hg} \)- Initial temperature at STP, \( T_1 = 273 \, \text{K} \) (or 0°C)- Final volume, \( V_2 = 350.0 \, \text{mL} \)- Final pressure, \( P_2 = 350 \, \text{mm Hg} \)We need to find the final temperature, \( T_2 \), in Celsius.

Apply Combined Gas Law

The combined gas law is given by the formula:\[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]We will use this relation to solve for \( T_2 \). Substitute the known values into the formula:\[\frac{760 \, \text{mm Hg} \times 50.0 \, \text{mL}}{273 \, \text{K}} = \frac{350 \, \text{mm Hg} \times 350.0 \, \text{mL}}{T_2}\]

Solve for Unknown Temperature

First, solve the equation for \( T_2 \). Cross-multiply to isolate \( T_2 \):\[T_2 = \frac{350 \, \text{mm Hg} \times 350.0 \, \text{mL} \times 273 \, \text{K}}{760 \, \text{mm Hg} \times 50.0 \, \text{mL}}\]Calculate:\[T_2 = \frac{33412500}{38000} = 879.28 \, \text{K}\]

Convert Kelvin to Celsius

Convert the final temperature from Kelvin to Celsius using the formula:\[T_{\text{Celsius}} = T_{\text{Kelvin}} - 273\]Thus:\[T_{\text{Celsius}} = 879.28 \, \text{K} - 273 = 606.28 \, ^{\circ}\text{C}\]

Key concepts

Ammonia Gas

Ammonia gas is a compound made of nitrogen and hydrogen with the formula NH₃. It is a colorless gas and has a distinctive, pungent smell. It is highly soluble in water and can be easily compressed into a liquid, making it a crucial component in many industrial and household applications. Ammonia is often used as a refrigerant in large industrial refrigeration systems and is also commonly used in the production of fertilizers. Additionally, it's present in various cleaning products due to its effectiveness in cutting through grime.
When studying the properties of ammonia gas, especially in a chemical context such as the combined gas law, it is important to understand that like many gases, ammonia will expand or contract with temperature and pressure changes. This makes ammonia a very suitable subject for problems related to gas law applications. In this specific exercise, we're looking at a scenario where the gas changes volume and pressure. By applying principles of the combined gas law, we can determine changes in other properties such as temperature, helping us understand the relationships between different state variables for ammonia gas.

STP Conditions

STP stands for Standard Temperature and Pressure. These are the conditions often used as a reference point in scientific experiments and calculations involving gases.

• The standard temperature is 0°C, which equates to 273 Kelvin (K).
• The standard pressure is 1 atmosphere, equivalent to 760 mm Hg or 101.325 kPa.

When gases are compared or calculated at STP, it's easier to predict their behavior since these conditions provide a baseline. In the context of the combined gas law, knowing the initial conditions at STP allows us to relate the starting conditions to those of a different state based on changes in volume or pressure. For example, in the given problem, the ammonia gas starts at STP and then changes as it undergoes variations in pressure and volume. By understanding and applying STP conditions, we can accurately determine how the other variables adjust and solve for unknowns, such as the final temperature after these changes.

Temperature Conversion

In the realm of gas law problems, converting between Kelvin and Celsius is a frequent necessity. The Kelvin scale is an absolute temperature scale commonly used in scientific calculations due to its direct relation to energy and particle movement.
For conversion, consider:

• Kelvin is the base unit where 0 K is absolute zero, the point at which particles contain minimum energy.
• To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature.

In this problem, once we determine the temperature using the combined gas law and solve for it in Kelvin, we then convert back to Celsius since it is easier to relate to in everyday scenarios. For instance, after calculating the temperature to be 879.28 K, converting to Celsius requires subtracting 273, resulting in approximately 606.28°C. This conversion is crucial in providing a clear understanding of the changes the ammonia gas undergoes between initial and final states as conditions vary.