Answer :
Sure, let's solve the problem step-by-step:
We are given:
- 2,000 BTUs added
- A volume of 1333 cubic feet (CF) of standard air
We need to find the final temperature in Fahrenheit after adding the 2,000 BTUs to the standard air.
First, let's look at the specific heat capacity of air and some key conversions we will need:
- The specific heat of air at constant pressure is approximately 1005 J/(kg·K).
- The density of standard air is about 1.225 kg/m³.
- 1 BTU is equivalent to about 1055.06 Joules.
- 1 cubic foot (CF) is equivalent to 0.0283168 cubic meters (m³).
### Conversion Steps
1. Convert the volume from cubic feet to cubic meters:
[tex]\[
\text{Volume in m}^3 = 1333 \, \text{CF} \times 0.0283168 \, \left(\frac{\text{m}^3}{\text{CF}}\right) = 37.7462944 \, \text{m}^3
\][/tex]
2. Calculate the mass of the air in kilograms:
[tex]\[
\text{Mass of air} = \text{Volume in m}^3 \times \text{Density of air} = 37.7462944 \, \text{m}^3 \times 1.225 \, \left(\frac{\text{kg}}{\text{m}^3}\right) = 46.23921064 \, \text{kg}
\][/tex]
3. Convert BTUs to Joules:
[tex]\[
\text{Energy added in Joules} = 2000 \, \text{BTUs} \times 1055.06 \, \left(\frac{\text{Joules}}{\text{BTU}}\right) = 2110120 \, \text{Joules}
\][/tex]
4. Calculate the temperature change (ΔT) in Kelvin using the formula [tex]\( Q = m \cdot c \cdot \Delta T \)[/tex]:
[tex]\[
\Delta T = \frac{\text{Energy added}}{\text{mass of air} \times \text{specific heat of air}} = \frac{2110120 \, \text{Joules}}{46.23921064 \, \text{kg} \times 1005 \, \left(\frac{\text{Joules}}{\text{kg} \cdot \text{K}}\right)} = 45.40782296 \, \text{K}
\][/tex]
5. Assume the initial temperature is 20 degrees Celsius (68 degrees Fahrenheit):
[tex]\[
\text{Initial temperature} = 20^\circ \text{C}
\][/tex]
6. Calculate the final temperature in Celsius:
[tex]\[
\text{Final temperature} = \text{Initial temperature} + \Delta T = 20^\circ \text{C} + 45.40782296^\circ \text{C} = 65.40782296^\circ \text{C}
\][/tex]
7. Convert the final temperature to Fahrenheit using the formula [tex]\( F = C \times \frac{9}{5} + 32 \)[/tex]:
[tex]\[
\text{Final temperature}^\circ \text{F} = 65.40782296^\circ \text{C} \times \frac{9}{5} + 32 = 150^\circ \text{F}
\][/tex]
So, the final temperature after adding 2,000 BTUs to 1333 CF of standard air is:
[tex]\[
\boxed{150^\circ \text{F}}
\][/tex]
Therefore, the correct choice is:
C. 151 degrees F
We are given:
- 2,000 BTUs added
- A volume of 1333 cubic feet (CF) of standard air
We need to find the final temperature in Fahrenheit after adding the 2,000 BTUs to the standard air.
First, let's look at the specific heat capacity of air and some key conversions we will need:
- The specific heat of air at constant pressure is approximately 1005 J/(kg·K).
- The density of standard air is about 1.225 kg/m³.
- 1 BTU is equivalent to about 1055.06 Joules.
- 1 cubic foot (CF) is equivalent to 0.0283168 cubic meters (m³).
### Conversion Steps
1. Convert the volume from cubic feet to cubic meters:
[tex]\[
\text{Volume in m}^3 = 1333 \, \text{CF} \times 0.0283168 \, \left(\frac{\text{m}^3}{\text{CF}}\right) = 37.7462944 \, \text{m}^3
\][/tex]
2. Calculate the mass of the air in kilograms:
[tex]\[
\text{Mass of air} = \text{Volume in m}^3 \times \text{Density of air} = 37.7462944 \, \text{m}^3 \times 1.225 \, \left(\frac{\text{kg}}{\text{m}^3}\right) = 46.23921064 \, \text{kg}
\][/tex]
3. Convert BTUs to Joules:
[tex]\[
\text{Energy added in Joules} = 2000 \, \text{BTUs} \times 1055.06 \, \left(\frac{\text{Joules}}{\text{BTU}}\right) = 2110120 \, \text{Joules}
\][/tex]
4. Calculate the temperature change (ΔT) in Kelvin using the formula [tex]\( Q = m \cdot c \cdot \Delta T \)[/tex]:
[tex]\[
\Delta T = \frac{\text{Energy added}}{\text{mass of air} \times \text{specific heat of air}} = \frac{2110120 \, \text{Joules}}{46.23921064 \, \text{kg} \times 1005 \, \left(\frac{\text{Joules}}{\text{kg} \cdot \text{K}}\right)} = 45.40782296 \, \text{K}
\][/tex]
5. Assume the initial temperature is 20 degrees Celsius (68 degrees Fahrenheit):
[tex]\[
\text{Initial temperature} = 20^\circ \text{C}
\][/tex]
6. Calculate the final temperature in Celsius:
[tex]\[
\text{Final temperature} = \text{Initial temperature} + \Delta T = 20^\circ \text{C} + 45.40782296^\circ \text{C} = 65.40782296^\circ \text{C}
\][/tex]
7. Convert the final temperature to Fahrenheit using the formula [tex]\( F = C \times \frac{9}{5} + 32 \)[/tex]:
[tex]\[
\text{Final temperature}^\circ \text{F} = 65.40782296^\circ \text{C} \times \frac{9}{5} + 32 = 150^\circ \text{F}
\][/tex]
So, the final temperature after adding 2,000 BTUs to 1333 CF of standard air is:
[tex]\[
\boxed{150^\circ \text{F}}
\][/tex]
Therefore, the correct choice is:
C. 151 degrees F
