Question

What mass of neon gas would be required to fill a 3.00 -L flask to a pressure of \(925 \mathrm{~mm} \mathrm{Hg}\) at 26 ' \(\mathrm{C}\) ? What mass of argon gas would be required to fill a similar flask to the same pressure at the same temperature?

Short answer

The mass of neon gas required to fill a 3.00 L flask at a pressure of 925 mmHg and 26°C is approximately \( m_{Ne} = 1.23 \mathrm{~g} \). The mass of argon gas required to fill a similar flask under the same conditions is approximately \( m_{Ar} = 2.30 \mathrm{~g} \).

Step-by-step solution

Convert given data to required units

First, we need to convert the given pressure and temperature to their respective units used in the ideal gas law equation: 1. Convert pressure from mmHg to atm: 1 atm = 760 mmHg, so 925 mmHg = \( \frac{925}{760} \) atm 2. Convert temperature from Celsius to Kelvin: T(K) = T(°C) + 273.15, so 26°C = 26 + 273.15 K

Calculate the number of moles of each gas using the ideal gas law equation

The ideal gas law equation is as follows: PV = nRT Where: P = pressure (in atm) V = volume (in L) n = number of moles (in mol) R = ideal gas constant (0.0821 L*atm/mol*K) T = temperature (in K) We know P, V, R, and T; we need to solve for n: n = \( \frac{PV}{RT} \)

Calculate the mass of Neon gas

To find the mass of neon gas (m_Ne), we need to use the molar mass of neon gas (M_Ne). m_Ne = n * M_Ne

Calculate the mass of Argon gas

To find the mass of argon gas (m_Ar), we need to use the molar mass of argon gas (M_Ar). m_Ar = n * M_Ar Now that we have the equations for the mass of both neon and argon gases, we can plug in the values:

Plugging in the values for Neon

1. P = \( \frac{925}{760} \) atm 2. V = 3.00 L 3. R = 0.0821 L*atm/mol*K 4. T = 26 + 273.15 K = 299.15 K 5. M_Ne = 20.18 g/mol n_Ne = \( \frac{PV}{RT} = \frac{( \frac{925}{760} * 3)}{ (0.0821 * 299.15)} \) m_Ne = n_Ne * M_Ne

Plugging in the values for Argon

1. P = \( \frac{925}{760} \) atm 2. V = 3.00 L 3. R = 0.0821 L*atm/mol*K 4. T = 26 + 273.15 K = 299.15 K 5. M_Ar = 39.95 g/mol n_Ar = \( \frac{PV}{RT} = \frac{( \frac{925}{760} * 3)}{ (0.0821 * 299.15)} \) m_Ar = n_Ar * M_Ar By calculating the values of m_Ne and m_Ar, we will get the required mass of neon and argon gases.

Key concepts

Converting Units in Gas Laws

Understanding how to convert units is crucial when dealing with gas laws. The ideal gas law, represented by the equation PV=nRT, requires specific units for each variable to ensure accuracy. Pressure (P) is often measured in atmospheres (atm), millimeters of mercury (mmHg), or kilopascals (kPa), while volume (V) is commonly given in liters (L). Temperature (T) needs to be in Kelvin (K), which can be converted from Celsius (°C) by adding 273.15. To convert pressure from mmHg to atm, you divide by 760, since 1 atm is equal to 760 mmHg. Remembering these conversion factors and consistently applying them will lead to the correct evaluation of the gas law equations.

For instance, in our example, we converted the given pressure of 925 mmHg to atm and temperature of 26°C to Kelvin by executing the following calculations:

• Pressure: \(925\,\text{mmHg} \times \frac{1\,\text{atm}}{760\,\text{mmHg}}\)
• Temperature: \(26°C + 273.15 = 299.15\,\text{K}\)

Such unit conversions are a fundamental step in solving gas law problems.

Calculating Molar Mass

Calculating the molar mass of a gas is essential when using the ideal gas law to find the mass of the gas. The molar mass (M) is the weight of one mole of a substance and is expressed in grams per mole (g/mol). It can be found by adding the atomic masses of all the atoms in a molecule of the substance, which are listed on the periodic table of elements. For monoatomic gases like neon (Ne) and argon (Ar), their molar masses are simply their atomic masses.

In our exercise, we use the molar masses of neon and argon to calculate the masses needed to fill the flasks:

• Molar mass of Neon (Ne): \(20.18\,\text{g/mol}\)
• Molar mass of Argon (Ar): \(39.95\,\text{g/mol}\)

By knowing the molar mass and the amount of moles (n), we can calculate the mass of the gases required.

Understanding PV=nRT

The Ideal Gas Law, PV=nRT, describes the relationship between the pressure (P), volume (V), temperature (T), and the number of moles (n) of an ideal gas. R is the ideal gas constant, which balances the units within the equation and has a value of 0.0821 L*atm/mol*K. This law assumes that gases are composed of many small particles that are far apart and move randomly, and the actual volume of the gas particles is much less than the space between them.

To solve for any one of the variables, you must know the other four. In our scenario, we are solving for n (the number of moles) with the equation rearranged to be \(n = \frac{PV}{RT}\). Once we determine the number of moles of gas, we can use this value to calculate the mass required to fill the flask.

Moles to Mass Conversion

Once we have computed the number of moles (n) of a gas, converting it to mass (m) involves using the molar mass (M). The equation for this conversion is simply \(m=nM\). It is a linear relationship where mass is directly proportional to the number of moles. When you've determined the number of moles using the ideal gas law, you multiply it by the molar mass of the gas to get the mass.

In our example, once the number of moles of neon (n_Ne) and argon (n_Ar) is calculated using the ideal gas law, the final step is to multiply by their respective molar masses to find the mass needed:

• Mass of Neon: \(m_{Ne} = n_{Ne} \times M_{Ne}\)
• Mass of Argon: \(m_{Ar} = n_{Ar} \times M_{Ar}\)

This straightforward multiplication will give us the mass in grams, the standard unit of mass in chemistry.