To determine the root-mean-square (rms) speed of an oxygen gas molecule at a temperature of [tex]\(31.0^\circ \text{C}\)[/tex], we can follow a series of steps. Let's go through each of these steps in detail:
### Step 1: Convert the temperature from Celsius to Kelvin
The first thing we need to do is convert the given temperature from degrees Celsius to Kelvin, since the Kelvin scale is the standard unit for temperature in physics equations involving gases.
The conversion formula is:
[tex]\[ \text{Temperature in Kelvin} (T_K) = \text{Temperature in Celsius} + 273.15 \][/tex]
Given:
[tex]\[ \text{Temperature in Celsius} = 31.0^\circ \text{C} \][/tex]
So,
[tex]\[ T_K = 31.0 + 273.15 = 304.15 \, \text{K} \][/tex]
### Step 2: Determine the molar mass of an oxygen molecule
The molecular weight of oxygen ([tex]\(O_2\)[/tex]) is given as 32.00 grams per mole. We convert this mass into kilograms per mole because the universal gas constant [tex]\(R\)[/tex] is generally expressed in terms of kilograms per mole.
[tex]\[ \text{Molar mass of } O_2 = \frac{32.00 \text{ g/mol}}{1000} = 0.032 \, \text{kg/mol} \][/tex]
### Step 3: Use the rms speed formula
The formula to calculate the rms speed ([tex]\(v_{\text{rms}}\)[/tex]) of a gas molecule is given by the equation:
[tex]\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \][/tex]
Where:
- [tex]\(R\)[/tex] is the universal gas constant, approximately [tex]\(8.314 \, \text{J}/(\text{mol} \cdot \text{K})\)[/tex]
- [tex]\(T\)[/tex] is the temperature in Kelvin
- [tex]\(M\)[/tex] is the molar mass in kilograms per mole
### Step 4: Plug in the values
Now we substitute the known values into the rms speed formula:
[tex]\[ R = 8.314 \, \text{J}/(\text{mol} \cdot \text{K}) \][/tex]
[tex]\[ T = 304.15 \, \text{K} \][/tex]
[tex]\[ M = 0.032 \, \text{kg/mol} \][/tex]
[tex]\[ v_{\text{rms}} = \sqrt{\frac{3 \times 8.314 \times 304.15}{0.032}} \][/tex]
### Step 5: Calculate the rms speed
Carrying out the calculation:
[tex]\[ v_{\text{rms}} \approx 486.89 \, \text{m/s} \][/tex]
So, the rms speed of an oxygen gas molecule at [tex]\(31.0^\circ \text{C}\)[/tex] is approximately [tex]\(486.89 \, \text{m/s}\)[/tex].
