9.4 kJ of energy is available to take a sample of ice initially at -40*C, heat it to it's melting point, melt it, and subsequently heat it to 50*C. What is the mass, in grams, of the largest such sample for which this amount of energy will suffice?

To solve this problem, we need to consider the energy required at each step of the process: heating the ice from -40°C to its melting point, melting the ice, and then heating the water from its melting point to 50°C. We can use the specific heat capacity and latent heat of fusion to determine the amount of energy required at each step.

1. Heating the ice from -40°C to its melting point:
The specific heat capacity of ice is 2.09 J/g°C. Since we have the initial temperature (-40°C) and the final temperature (0°C, the melting point of ice), we can calculate the energy required to heat the ice using the formula:

Q = mcΔT

Where:
Q is the energy in Joules
m is the mass in grams
c is the specific heat capacity in J/g°C
ΔT is the change in temperature in °C

ΔT = (0°C) - (-40°C) = 40°C

Q = mcΔT = (m)(2.09 J/g°C)(40°C)

2. Melting the ice:
The latent heat of fusion of ice is 334 J/g. To calculate the energy required to melt the ice, we can use the formula:

Q = mL

Where:
Q is the energy in Joules
m is the mass in grams
L is the latent heat of fusion in J/g

Q = mL = (m)(334 J/g)

3. Heating the water from its melting point to 50°C:
The specific heat capacity of water is 4.18 J/g°C. Similar to step 1, we can calculate the energy required using the formula:

Q = mcΔT

Where:
Q is the energy in Joules
m is the mass in grams
c is the specific heat capacity in J/g°C
ΔT is the change in temperature in °C

ΔT = 50°C - 0°C = 50°C

Q = mcΔT = (m)(4.18 J/g°C)(50°C)

Now, we can sum up the energy required at each step and set it equal to the available energy of 9.4 kJ (9.4 × 10^3 J). Rearranging the equation and solving for mass (m), we can find the maximum mass of the ice:

(m)(2.09 J/g°C)(40°C) + (m)(334 J/g) + (m)(4.18 J/g°C)(50°C) = 9.4 × 10^3 J

Solve this equation for the value of m, which will give you the maximum mass of the ice sample in grams that the given energy will suffice.