Problem 93 - Chemistry Matter and Change

Question

Television Determine the pressure inside a television picture tube with a volume of 3.50 \(L\) that contains \(2.00 \times 10^{-5}\) g of nitrogen gas at \(22.0^{\circ} C .\)

Step-by-Step Solution

Answer

The pressure inside the television picture tube is approximately \(5.24 \times 10^{-5}\) atmospheres.

Step 1: Convert the mass of nitrogen gas to moles

To convert grams to moles, we'll use the molar mass of nitrogen, which is approximately \(14.01\, \frac{g}{mol}\) for each Nitrogen atom. Since nitrogen gas is diatomic (N\(_2\)), the molar mass is \(2 \times 14.01 = 28.02\, \frac{g}{mol}\). To find the number of moles: \(n = \frac{m}{M}\) Where: - \(n\) is the number of moles - \(m\) is the mass of the gas - \(M\) is the molar mass \(n = \frac{2.00 \times 10^{-5}\, g}{28.02\, \frac{g}{mol}}\)

Step 2: Calculate the number of moles

Now, we can calculate the number of moles of nitrogen gas: \(n = \frac{2.00 \times 10^{-5}\, g}{28.02\, \frac{g}{mol}} = 7.14 \times 10^{-7}\, mol\)

Step 3: Convert the temperature to Kelvin

Next, we need to convert the given temperature from Celsius to Kelvin. To do this, we can use the following equation: \(T (K) = T (°C) + 273.15\) Now, let's convert the given temperature: \(T (K) = 22.0^{\circ} C + 273.15 = 295.15\, K\)

Step 4: Calculate the pressure using the Ideal Gas Law

We have all the values needed to solve the Ideal Gas Law equation: \(PV = nRT\) Where: - P is the pressure (our unknown value) - V is the volume of the tube, \(3.50 L\) - n is the number of moles, \(7.14 \times 10^{-7}\, mol\) - R is the ideal gas constant, \(0.0821\, \frac{L \cdot atm}{mol \cdot K}\) - T is the temperature in Kelvin, \(295.15\, K\) Now, let's solve for the pressure: \(P = \frac{nRT}{V} = \frac{(7.14 \times 10^{-7}\, mol)(0.0821 \frac{L \cdot atm}{mol \cdot K})(295.15\, K)}{3.50\, L}\)

Step 5: Determine the pressure inside the television picture tube

Finally, we can calculate the pressure inside the television picture tube: \(P = \frac{(7.14 \times 10^{-7}\, mol)(0.0821\, \frac{L \cdot atm}{mol \cdot K})(295.15\, K)}{3.50\, L} = 5.24 \times 10^{-5}\, atm\) Thus, the pressure inside the television picture tube is approximately \(5.24 \times 10^{-5}\) atmospheres.

Key Concepts

Pressure Calculation

Pressure calculation is a key application of the Ideal Gas Law. This law builds a relationship between four main variables associated with gases: Pressure (P), Volume (V), Number of Moles (n), and Temperature (T). In this exercise, we uncover how these variables can be used to determine the pressure inside a television picture tube filled with nitrogen gas.
Using the Ideal Gas Law, represented as \( PV = nRT \), we aim to find the pressure \( P \) within the tube. Here is a step-by-step breakdown:

• First, you need to have precise values for the number of moles \( n \), the temperature \( T \) in Kelvin, the volume \( V \) of the container, and the ideal gas constant \( R = 0.0821 \, \frac{L \cdot atm}{mol \cdot K} \).
• The volume \( V \) is given as 3.50 liters.
• Make sure the other values are appropriately calculated, especially the number of moles and temperature, which need conversions.

Remember to plug the values into the equation and solve for \( P \), giving you the pressure within the television picture tube. This basic formula helps solve various real-world and theoretical gas-related problems.

Mole Conversion

Converting mass to moles is an essential skill in chemistry, particularly when working with gas laws. Mole conversion gives us a pathway to link the given mass of a substance to its amount in moles, a standard unit in chemistry.

In this exercise, we convert the mass of nitrogen gas into moles using its molar mass. Consider the following critical points:

• Recall that nitrogen gas exists as a diatomic molecule \( (N_2) \). So, we need to adjust the molar mass for \( N_2 \), which is approximately \( 28.02 \, \frac{g}{mol} \).
• Apply the conversion formula \( n = \frac{m}{M} \), where \( m \) is the mass of the gas, and \( M \) is its molar mass.
• In this scenario, \( m = 2.00 \times 10^{-5} \, g \), leading to \( n = 7.14 \times 10^{-7} \, mol \) of nitrogen gas.

This conversion ensures we have the correct number of moles to use in the Ideal Gas Law equation, facilitating an accurate calculation of the pressure in the television tube.

Temperature Conversion

Converting temperature from Celsius to Kelvin is a vital step when utilizing the Ideal Gas Law. The Kelvin scale is the standard in thermodynamic calculations because it starts at absolute zero, ensuring no negative values for temperature, which is compatible with energy calculations.

Here’s how to handle the conversion effectively:

• Use the formula \( T (K) = T (°C) + 273.15 \) to convert temperatures from Celsius to Kelvin.
• This exercise gives us a temperature of \( 22.0^{\circ} C \), which can be converted to \( 295.15\, K \).
• Make sure the temperature is always calculated in Kelvin before substituting into the Ideal Gas Law. This precision ensures the consistency of units and the accuracy of the calculated pressure or other gas properties.

Understanding temperature conversion is crucial. It ensures the measurements fit well in the universal framework that physicists and chemists use for calculations on gases.