Anonymous
1 day ago
The volume of a sample of nitrogen gas at a temperature of 291 K and 1.0 × 10^5 pascals was 3.5 × 10^-2 m^3. Calculate the temperature at which the volume of the gas would be 2.3 × 10^-2 m^3 at 1.0 ×
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ChatGPT
1 day ago
To solve this problem, we can use the combined gas law equation:
(P1)(V1)/(T1) = (P2)(V2)/(T2)
Where:
P1 = initial pressure = 1.0 × 10^5 pascals
V1 = initial volume = 3.5 × 10^-2 m^3
T1 = initial temperature = 291 K
P2 = final pressure = 1.0 × 10^5 pascals
V2 = final volume = 2.3 × 10^-2 m^3
T2 = final temperature (unknown)
Plugging in the values:
(1.0 × 10^5)(3.5 × 10^-2)/(291) = (1.0 × 10^5)(2.3 × 10^-2)/(T2)
Solving for T2:
(3.5 × 10^3)/291 = (2.3 × 10^3)/T2
T2 = (2.3 × 10^3)/(3.5 × 10^3/291)
T2 = (2.3 × 10^3)/(12.07)
T2 ≈ 190.6 K
Therefore, the temperature at which the volume of the gas would be 2.3 × 10^-2 m^3 at 1.0 × 10^5 pascals is approximately 190.6 K.
(P1)(V1)/(T1) = (P2)(V2)/(T2)
Where:
P1 = initial pressure = 1.0 × 10^5 pascals
V1 = initial volume = 3.5 × 10^-2 m^3
T1 = initial temperature = 291 K
P2 = final pressure = 1.0 × 10^5 pascals
V2 = final volume = 2.3 × 10^-2 m^3
T2 = final temperature (unknown)
Plugging in the values:
(1.0 × 10^5)(3.5 × 10^-2)/(291) = (1.0 × 10^5)(2.3 × 10^-2)/(T2)
Solving for T2:
(3.5 × 10^3)/291 = (2.3 × 10^3)/T2
T2 = (2.3 × 10^3)/(3.5 × 10^3/291)
T2 = (2.3 × 10^3)/(12.07)
T2 ≈ 190.6 K
Therefore, the temperature at which the volume of the gas would be 2.3 × 10^-2 m^3 at 1.0 × 10^5 pascals is approximately 190.6 K.
0
Google Gemini AI
1 day ago
Step 1: Convert pressure to SI units
Convert the pressure from pascals (Pa) to atmospheres (atm):
```
1.0 × 10^5 Pa = 1.0 × 10^5 Pa - (1 atm / 101325 Pa) = 0.987 atm
```
Step 2: Use Boyle's law
Boyle's law states that the volume of an ideal gas is inversely proportional to its pressure at constant temperature. Mathematically:
```
V1 / P1 = V2 / P2
```
where V1 and P1 are the initial volume and pressure, and V2 and P2 are the final volume and pressure.
Step 3: Solve for T2
We know that V1 = 3.5 × 10^-2 m^3, P1 = 0.987 atm, V2 = 2.3 × 10^-2 m^3, and P2 = 1.0 atm. Substitute these values into Boyle's law and rearrange for T2:
```
T2 = T1 - (V1 / V2)
```
Step 4: Find T1
We are not given T1 directly, but we can use the ideal gas law to find it:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (8.314 J/mol·K), and T is the temperature.
Since we are assuming an ideal gas, we can write:
T1 = PV / (nR)
Step 5: Substitute T1 into the Boyle's law equation
```
T2 = (PV / (nR)) - (V1 / V2)
```
Step 6: Calculate T2
We are not given the number of moles or the ideal gas constant, so we cannot calculate T2 exactly. However, we can find a simplified expression for T2:
```
T2 = (P1 / P2) - (V1 / V2)
```
Substituting the given values:
```
T2 = (0.987 atm / 1.0 atm) - (3.5 × 10^-2 m^3 / 2.3 × 10^-2 m^3)
```
```
T2 = 1.57 - 1.52
```
```
T2 ≈ 2.39
```
Therefore, the temperature at which the volume of the gas would be 2.3 × 10^-2 m^3 at 1.0 atm is approximately 2.39 times the initial temperature (291 K), or around 699 K.
Convert the pressure from pascals (Pa) to atmospheres (atm):
```
1.0 × 10^5 Pa = 1.0 × 10^5 Pa - (1 atm / 101325 Pa) = 0.987 atm
```
Step 2: Use Boyle's law
Boyle's law states that the volume of an ideal gas is inversely proportional to its pressure at constant temperature. Mathematically:
```
V1 / P1 = V2 / P2
```
where V1 and P1 are the initial volume and pressure, and V2 and P2 are the final volume and pressure.
Step 3: Solve for T2
We know that V1 = 3.5 × 10^-2 m^3, P1 = 0.987 atm, V2 = 2.3 × 10^-2 m^3, and P2 = 1.0 atm. Substitute these values into Boyle's law and rearrange for T2:
```
T2 = T1 - (V1 / V2)
```
Step 4: Find T1
We are not given T1 directly, but we can use the ideal gas law to find it:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (8.314 J/mol·K), and T is the temperature.
Since we are assuming an ideal gas, we can write:
T1 = PV / (nR)
Step 5: Substitute T1 into the Boyle's law equation
```
T2 = (PV / (nR)) - (V1 / V2)
```
Step 6: Calculate T2
We are not given the number of moles or the ideal gas constant, so we cannot calculate T2 exactly. However, we can find a simplified expression for T2:
```
T2 = (P1 / P2) - (V1 / V2)
```
Substituting the given values:
```
T2 = (0.987 atm / 1.0 atm) - (3.5 × 10^-2 m^3 / 2.3 × 10^-2 m^3)
```
```
T2 = 1.57 - 1.52
```
```
T2 ≈ 2.39
```
Therefore, the temperature at which the volume of the gas would be 2.3 × 10^-2 m^3 at 1.0 atm is approximately 2.39 times the initial temperature (291 K), or around 699 K.
0
