Assume that carbon dioxide in a 2.00 L soda bottle is dissolved at a pressure of 2.07 bar. At 2.28°C, the Henry’s law constant for carbon dioxide dissolved in water is 0.068 L/bar. Calculate the concentration of dissolved carbon dioxide in mol/L and the mass of carbon dioxide that can be dissolved in the bottle of soda at the given pressure.
The Correct Answer and Explanation is:1
Here is a step-by-step solution to the problem presented in the image.
Given Information:
- Volume of soda (V) = 2.00 L
- Pressure of CO₂ (P_CO₂) = 2.07 bar
- Henry’s Law constant (k_H) = 0.068 mol / (L·bar)
- The problem asks for the concentration (in mol/L) and the mass (in grams) of the dissolved carbon dioxide.
Part 1: Calculate the concentration of dissolved CO₂
We will provided.
This problem uses Henry’s Law, which states that the concentration of a dissolved gas in a liquid is directly proportional to the partial pressure of that gas above the liquid.
Given Information:
- Volume of soda (V): 2.00 L
- Pressure of CO₂ (P): 2.07 bar
- Henry’s Law Constant (k_H): 0.068 mol / (L·bar)
- Molar Mass of CO₂ (M):
- C = 12.01 g/mol
- O = 16.00 g/mol
- M(CO₂) = 12.01 + 2(16.00) = 44.01 g/mol
Part 1: Calculate the Concentration of Dissolved CO₂
The formula for Henry’s Law is:
C = k_H × P
Where:
- C is the concentration in mol/L
- k_H is the Henry’s Law constant
- P is the partial pressure of the gas
Calculation:
C = (0.068 mol / (L·bar)) × (2.07 bar)
C = 0.14076 mol/L
Rounding to the correct number of significant figures (2, limited by the constant 0.068):
C ≈ 0.14 mol/L
The concentration of dissolved carbon dioxide is 0.14 mol/L.
Part 2: Calculate the Mass of Dissolved CO₂
First, we need to find the total number of moles (n) of CO₂ dissolved in the 2.00 L bottle using the concentration we just calculated.
Formula:
n = C × V
Calculation (using the unrounded concentration for accuracy):
n = (0.14076 mol/L) × (2.00 L)
n = 0.28152 mol
Next use Henry’s Law, which states that the concentration (C) of a dissolved gas is directly proportional to the partial pressure (P) of that gas above the solution.
Formula:
C = k_H × P
Calculation:
Substitute the given values into the formula:
C = (0.068 mol / (L·bar)) × (2.07 bar)
C = 0.14076 mol/L
Significant Figures:
The Henry’s law constant (0.068) has two significant figures, and the pressure (2.07) has three. The result should be rounded to the least number of significant figures, which is two.
Concentration = 0.14 mol/L
Part 2: Calculate the mass of dissolved CO₂
First, we need to find the total number of moles (n) of CO₂ dissolved in the 2.00 L bottle. Then, we can convert moles to mass using the molar mass of CO₂.
Step 2a: Calculate the moles of CO₂
Formula:
Moles (n) = Concentration (C) × Volume (, we convert the moles of CO₂ to mass (m) using its molar mass.
Formula:
m = n × M
Calculation:
m = (0.28152 mol) × (44.01 g/mol )
m = 12.3896 g
Rounding to the correct number of significant figures (2, again limited by the constant 0.068):
m ≈ 12 g
**The mass of carbon dioxide that can be dissolved in the bottle is 12 g.**V)
Calculation:
Using the unrounded concentration from the previous step to maintain accuracy:
n = (0.14076 mol/L) × (2.00 L)
n = 0.28152 mol
Step 2b: Calculate the mass of CO₂
Formula:
Mass (m) = Moles (n) × Molar Mass (M)
Molar Mass of CO₂:
- Carbon (C) ≈ 12.01 g/mol
- Oxygen (O) ≈ 16.00 g/mol
- M_CO₂ = 12.01 + 2(16.00) = 44.01 g/mol
Calculation:
m = (0.28152 mol) × (44.01 g/mol )
m = 12.3896 g
Significant Figures:
The calculation involves values with two (from k_H), three (from V), and three (from P) significant figures. The final answer must be rounded to two significant figures.
Mass = 12 g
Final Answer:
- The concentration of dissolved carbon dioxide is 0.14 mol/L.
- The mass of carbon dioxide that can be dissolved in the bottle is 12 g.
