To determine the mass of helium involved when it is heated from [tex]\(20^{\circ} C\)[/tex] to [tex]\(100^{\circ} C\)[/tex] with 520 MJ of energy, we can use the formula that relates energy, mass, specific heat capacity, and temperature change. The formula is:
[tex]\[ Q = m \cdot c \cdot \Delta T \][/tex]
where:
- [tex]\( Q \)[/tex] is the energy transferred (in joules),
- [tex]\( m \)[/tex] is the mass of helium (in kilograms),
- [tex]\( c \)[/tex] is the specific heat capacity (in joules per kilogram per degree Celsius),
- [tex]\( \Delta T \)[/tex] is the change in temperature (in degrees Celsius).
Let's break down the steps to solve for the mass [tex]\( m \)[/tex]:
1. Convert the energy transferred from megajoules to joules:
Since 1 MJ (megajoule) is equal to [tex]\( 10^6 \)[/tex] joules, you need to convert 520 MJ to joules:
[tex]\[ Q = 520 \times 10^6 \, \text{J} \][/tex]
2. Calculate the change in temperature ([tex]\( \Delta T \)[/tex]):
The initial temperature ([tex]\( T_i \)[/tex]) is [tex]\(20^{\circ} C\)[/tex] and the final temperature ([tex]\( T_f \)[/tex]) is [tex]\(100^{\circ} C\)[/tex]:
[tex]\[ \Delta T = T_f - T_i \][/tex]
[tex]\[ \Delta T = 100^{\circ} C - 20^{\circ} C \][/tex]
[tex]\[ \Delta T = 80^{\circ} C \][/tex]
3. Rearrange the energy formula to solve for mass ([tex]\( m \)[/tex]):
[tex]\[ m = \frac{Q}{c \cdot \Delta T} \][/tex]
4. Substitute the given values into the rearranged formula:
Given that [tex]\( Q = 520 \times 10^6 \)[/tex] J, [tex]\( c = 5200 \,\text{J/kg} \cdot ^\circ C \)[/tex], and [tex]\( \Delta T = 80^{\circ} C\)[/tex]:
[tex]\[ m = \frac{520 \times 10^6 \, \text{J}}{5200 \, \text{J/kg} \cdot ^\circ C \times 80^{\circ} C} \][/tex]
5. Perform the calculation:
[tex]\[ m = \frac{520 \times 10^6}{5200 \times 80} \][/tex]
[tex]\[ m = \frac{520 \times 10^6}{416000} \][/tex]
[tex]\[ m = 1250 \, \text{kg} \][/tex]
Therefore, the mass of helium involved is [tex]\( 1250 \)[/tex] kg.
