Answer :
To determine the volume required to hold 2.0 moles of gas, given that 0.50 moles occupy 10.5 liters, we can use the relationship for gases under constant temperature and pressure. This relationship is derived from the Ideal Gas Law and suggests that the volume of a gas is directly proportional to the number of moles, assuming other conditions (pressure and temperature) remain constant.
### Step-by-Step Solution:
1. Identify Known Quantities:
- Initial number of moles, [tex]\( n_1 = 0.50 \)[/tex] moles
- Initial volume, [tex]\( V_1 = 10.5 \)[/tex] liters
- Final number of moles, [tex]\( n_2 = 2.0 \)[/tex] moles
2. Set Up the Proportional Relationship:
The relationship between the volumes and the number of moles can be expressed as:
[tex]\[ \frac{V_1}{n_1} = \frac{V_2}{n_2} \][/tex]
This equation states that the ratio of volume to the number of moles remains constant.
3. Rearrange the Equation to Solve for [tex]\( V_2 \)[/tex] (the final volume):
[tex]\[ V_2 = V_1 \times \frac{n_2}{n_1} \][/tex]
4. Substitute the Known Values into the Equation:
[tex]\[ V_2 = 10.5 \times \frac{2.0}{0.50} \][/tex]
5. Simplify the Calculation:
[tex]\[ V_2 = 10.5 \times 4 \][/tex]
[tex]\[ V_2 = 42.0 \][/tex]
### Conclusion:
The volume required to hold 2.0 moles of gas, under the same conditions of pressure and temperature, is:
[tex]\[ V_2 = 42.0 \text{ liters} \][/tex]
### Step-by-Step Solution:
1. Identify Known Quantities:
- Initial number of moles, [tex]\( n_1 = 0.50 \)[/tex] moles
- Initial volume, [tex]\( V_1 = 10.5 \)[/tex] liters
- Final number of moles, [tex]\( n_2 = 2.0 \)[/tex] moles
2. Set Up the Proportional Relationship:
The relationship between the volumes and the number of moles can be expressed as:
[tex]\[ \frac{V_1}{n_1} = \frac{V_2}{n_2} \][/tex]
This equation states that the ratio of volume to the number of moles remains constant.
3. Rearrange the Equation to Solve for [tex]\( V_2 \)[/tex] (the final volume):
[tex]\[ V_2 = V_1 \times \frac{n_2}{n_1} \][/tex]
4. Substitute the Known Values into the Equation:
[tex]\[ V_2 = 10.5 \times \frac{2.0}{0.50} \][/tex]
5. Simplify the Calculation:
[tex]\[ V_2 = 10.5 \times 4 \][/tex]
[tex]\[ V_2 = 42.0 \][/tex]
### Conclusion:
The volume required to hold 2.0 moles of gas, under the same conditions of pressure and temperature, is:
[tex]\[ V_2 = 42.0 \text{ liters} \][/tex]
