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How many moles of oxygen (O₂) are contained in a 3.6 L cylinder that has a pressure of 2601 mmHg and a temperature of 31°C? Be sure your answer has the correct number of significant figures.

(Note: Reference the Fundamental constants and Conversion factors for non-SI units tables for additional information.)

Sagot :

GinnyAnsweridn
To determine the number of moles of oxygen (O₂) contained in a 3.6 L cylinder with a pressure of 2601 mmHg and a temperature of 31°C, we can use the Ideal Gas Law equation, \( PV = nRT \), where:

- \( P \) is pressure
- \( V \) is volume
- \( n \) is the number of moles
- \( R \) is the universal gas constant
- \( T \) is temperature in Kelvin

Here’s a step-by-step solution:

Step 1: Convert Temperature to Kelvin

We are given the temperature in Celsius (31°C), which we need to convert to Kelvin. The conversion formula is:

[tex]\[ T(K) = T(°C) + 273.15 \][/tex]

So,

[tex]\[ T(K) = 31 + 273.15 = 304.15 \, K \][/tex]

Step 2: Convert Pressure to atm

We are given the pressure in mmHg (2601 mmHg), which we need to convert to atmospheres (atm). The conversion factor is 1 atm = 760 mmHg. The conversion formula is:

[tex]\[ P(\text{atm}) = \frac{P(\text{mmHg})}{760} \][/tex]

So,

[tex]\[ P(\text{atm}) = \frac{2601}{760} = 3.422368421 \, atm \][/tex]

Step 3: Use the Ideal Gas Law to Solve for n (number of moles)

[tex]\[ PV = nRT \][/tex]

We can rearrange the equation to solve for \( n \):

[tex]\[ n = \frac{PV}{RT} \][/tex]

Substitute the values:

- \( P = 3.422368421 \, atm \)
- \( V = 3.6 \, L \)
- \( R = 0.0821 \, \text{L} \cdot \text{atm} / (\text{mol} \cdot \text{K}) \)
- \( T = 304.15 \, K \)

[tex]\[ n = \frac{(3.422368421 \, atm \times 3.6 \, L)}{(0.0821 \, \text{L} \cdot \text{atm} / (\text{mol} \cdot \text{K}) \times 304.15 \, K)} \][/tex]

[tex]\[ n = \frac{12.3205263156 \, \text{L} \cdot \text{atm}}{24.962215 \, \text{L} \cdot \text{atm/mole}} \][/tex]

[tex]\[ n = 0.4933990202 \, \text{moles} \][/tex]

Step 4: Round to Appropriate Significant Figures

Given the significant figures in the problem's measurements (2601 mmHg with four significant digits, 3.6 L with two significant digits, and 31°C with two significant digits), the final answer should be rounded to two significant digits.

Therefore, the number of moles of oxygen is:

[tex]\[ n ≈ 0.49 \, \text{moles} \][/tex]

So, the number of moles of oxygen (O₂) contained in the cylinder is approximately 0.49 moles.
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