For his 18th birthday in February Peter plants to turn a hut in the garden of his parents into a swimming pool with an artifical beach. In order to estimate the consts for heating the water and the house , peter obtains the data for the natural gas combustion and its price.
What is the total energy (in MJ) needed for Peter's "winter swimming pool" calculated in 1.3 and 1.4? How much natural gas will he need, if the gas heater has an efficiency of 90.0% ?
What are the different costs for the use of either natural gas or electricity ? Use the values given by PUC for your calculations and assume 100% efficiency for the electric heater.
Table 1: Composition of natural gas
`{:("Chemical substance","mol fraction x",D_(1)H^(@)(KJ mol^(-1))^(-1),S^(@)(J mol^(-1)K^(-1))^(-1),C_(p)^(@)(J mol^(-1)K^(-1))^(-1)),(CO_(2(g)),0.0024,-393.5,213.6,37.1),(N_(2(g)) ,0.0134,0.0,191.6,29.1),(CH_(2(g)),0.9732,-74.6,186.3,35.7),(C_(2)H_(3 (g)),0.0110,-84.0,229.2,52.2),(H_(2)O_(g),-,-285.8,70.0,75.3),(H_(2)O_(g),-,-241.8,188.8,33.6),(H_(2)O_(g),-,0.0,205.2,29.4):}`
Equation
`J=E(A.Deltat)^(-1) =!! lambda "wall" . DeltaT. d^(-1)`, where
J= energy flow E along a temperature gradient (wall direction Z) par area A and time `Deltat`, d-wall thickness , `lambda`wall -heat conductivity , `DeltaT` - difference in temperature between the inside and the outside of the house.

Answer

To solve the problem, we need to calculate the total energy required for heating the water and the house, determine the amount of natural gas needed, and then compare the costs of using natural gas versus electricity.

### Step 1: Calculate the total energy needed for heating
1. **Identify the parameters**: We need the volume of water to be heated, the specific heat capacity of water, and the temperature difference (ΔT) between the initial and final temperatures.
2. **Use the formula**:
\[ Q = m \cdot c \cdot \Delta T \]
where \( Q \) is the heat energy (in Joules), \( m \) is the mass of water (in kg), \( c \) is the specific heat capacity of water (approximately 4.18 kJ/kg·K), and \( \Delta T \) is the temperature change (in °C or K).

### Step 2: Convert energy to MJ
1. **Convert Joules to Megajoules**: Since 1 MJ = 1,000,000 J, we will convert the total energy calculated in step 1 to MJ.

### Step 3: Calculate the amount of natural gas required
1. **Determine the energy content of natural gas**: From the data provided, we need to find the enthalpy of combustion for the main component (CH₄) of natural gas.
2. **Calculate the energy required from natural gas**:
\[ \text{Energy required from natural gas} = \frac{Q}{\text{Efficiency}} \]
where Efficiency = 0.90 (90% efficiency of the gas heater).
3. **Convert this energy to moles of natural gas** using the enthalpy of combustion.

### Step 4: Calculate costs for natural gas and electricity
1. **Determine the cost of natural gas**: Multiply the amount of natural gas needed (in moles) by the price per mole or per MJ.
2. **Calculate the cost of electricity**: Since the electric heater is assumed to have 100% efficiency, the total energy required (in MJ) will be the same as calculated in step 2. Multiply this energy by the cost per MJ of electricity.

### Step 5: Compare the costs
1. **Comparison**: Compare the total costs of using natural gas and electricity to determine which is more economical.