Question

Calculate the number of moles of \(O^{2}\) gas held in a sealed, \(2.00-L\) tank at 3.50 \(\mathrm{atm}\) and \(25.0^{\circ} C .\) How many moles would be in the tank if the temperature was raised to \(49.0^{\circ} C\) and the pressure remained constant?

Short answer

There are approximately 0.286 moles of \(O_2\) gas at 25.0 °C in the tank. When the temperature is raised to 49.0 °C and the pressure remains constant, there would be approximately 0.313 moles of \(O_2\) gas in the tank.

Step-by-step solution

Convert given temperature to Kelvin

Before using the Ideal Gas Law formula, we need to convert the given temperatures from Celsius to Kelvin. The formula for this conversion is: \(K = °C + 273.15\) At 25.0 °C: \(T_1 = 25.0 + 273.15 = 298.15 K\) At 49.0 °C: \(T_2 = 49.0 + 273.15 = 322.15 K\)

Calculate the number of moles (n) at 25.0 °C

To calculate the number of moles, we will now use the Ideal Gas Law formula and rearrange it to solve for 'n': \(n = \frac{PV}{RT}\) We're given the pressure (3.50 atm), volume (2.00 L), and we have already calculated the temperature in Kelvin (298.15 K). We'll use the value of the gas constant R = 0.0821 L atm/mol K (as our pressure is in atm and volume in L). Now we can plug in our values: \(n_1 = \frac{ (3.50\,\mathrm{atm})(2.00\,\mathrm{L}) }{ (0.0821\, \mathrm{L\,atm/mol\,K})(298.15\,\mathrm{K}) }\) \(n_1 = \frac{7}{24.46}\) \(n_1 ≈ 0.286\, moles\) So, we have approximately 0.286 moles of \(O_2\) gas at 25.0 °C.

Calculate the number of moles (n) at 49.0 °C with constant pressure

Since we are given that the pressure remains constant, we can use the combined gas law, which states: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) But, as moles and pressure are constant, \(V_1\,=\,V_{2}\) Therefore, we can write: \(\frac{n_1}{T_1} = \frac{n_2}{T_2}\) Now, we can plug in our values to find the number of moles at 49.0 °C: \(n_2 = n_1 \times\frac{ T_2}{ T_1}\) \(n_2 = 0.286 \times \frac{322.15\,\mathrm{K}}{298.15\,\mathrm{K}}\) \(n_2 ≈ 0.313\, moles\) So, there would be approximately 0.313 moles of \(O_2\) gas when the temperature is raised to 49.0 °C.

Key concepts

Gas Constant

The gas constant, symbolized as R, is a pivotal value in various gas-related calculations. It features in the Ideal Gas Law, which relates the pressure, volume, temperature, and mole number of a gas. R is essentially the factor that converts units of temperature and volume to units of energy, allowing us to relate the macroscopic properties of gases in a consistent way. In this context, we use R with a value of 0.0821 L atm/mol K because the pressure is measured in atmospheres (atm) and volume in liters (L). This makes calculations harmonious when employing the Ideal Gas Law. The universality of R empowers it to bridge diverse unit systems, making it a fundamental part of thermodynamics and chemistry.

Convert Celsius to Kelvin

Converting Celsius to Kelvin is a fundamental step in gas calculations, because gas laws require temperature to be in the Kelvin scale. The Kelvin scale is an absolute temperature scale, starting at absolute zero, unlike the Celsius scale, which is relative to the freezing point of water. The relationship between Celsius and Kelvin is linear, with a direct conversion formula: Kelvin = Celsius + 273.15. Although sometimes approximated as 273 for simplicity, using the exact value ensures precision in scientific calculations. The necessity of this conversion lies in the nature of temperature as a measure of kinetic energy in the particles of a substance. In the Kelvin scale, zero really means zero kinetic energy, which aligns with the principles of thermodynamics.

Combined Gas Law

The combined gas law is an amalgamation of the three empirical gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law. It succinctly relates the pressure, volume, and temperature of a gas, provided the amount of gas remains constant. In its form, the combined gas law is expressed as:
\(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\).
When conditions change, like in the exercise where the temperature increases while the pressure remains stable, the combined gas law becomes critical in determining the new state of the gas. By manipulating the combined gas law's formula, it is possible to hold different variables constant as needed, illustrating the law's versatility in different scenarios involving gas conditions.

Calculate Moles of Gas

Calculating moles of gas is often required to understand the amount of a substance present in a given volume at a particular temperature and pressure. By using the Ideal Gas Law, we can solve for the number of moles (n) by rearranging the equation to \(n = \frac{PV}{RT}\). Here, P represents pressure, V is volume, T is temperature, and R is the ideal gas constant. The mole is a standard unit of quantity in chemistry, allowing scientists to count particles of a substance based on a common reference - the number of atoms in 12 grams of carbon-12, which is Avogadro's number. Calculating moles becomes crucial in chemical reactions, stoichiometry, and when comparing the amount of substances in different conditions, as demonstrated in the exercise provided.