To determine the appropriate volume ratio of additive X and water for the radiator fluid in an automobile operating in extreme cold conditions, we need to consider the freezing point depression that occurs when a solute is added to a solvent. In this case, we want to ensure that the mixture remains liquid at temperatures as low as -40 degrees Celsius.
Understanding Freezing Point Depression
Freezing point depression is a colligative property, which means it depends on the number of solute particles in a solution rather than the identity of the solute. The formula for calculating the freezing point depression is:
ΔT_f = k_f * m
Where:
- ΔT_f is the change in freezing point (in Kelvin).
- k_f is the freezing point depression constant of the solvent (water in this case).
- m is the molality of the solution (moles of solute per kilogram of solvent).
Calculating the Required Freezing Point Depression
Given that the maximum drop in temperature is -40 degrees Celsius, we need to calculate the freezing point depression:
ΔT_f = 0 - (-40) = 40 degrees Celsius
Finding the Molality
Next, we can use the freezing point depression constant for water, which is given as k_f = 2 K kg/mol. We can rearrange the freezing point depression formula to find the molality:
m = ΔT_f / k_f = 40 / 2 = 20 mol/kg
Calculating the Amount of Additive X
Now that we have the molality, we can determine how many moles of additive X we need. Since molality is defined as moles of solute per kilogram of solvent, we need to consider 1 kg of water (since the density of water is 1 g/ml, 1 kg equals 1000 ml).
To find the number of moles of X required:
moles of X = molality * mass of water (kg) = 20 mol/kg * 1 kg = 20 moles
Calculating the Mass of Additive X
Now, we can convert moles of X to grams using its molecular weight:
mass of X = moles * molecular weight = 20 moles * 62 g/mol = 1240 grams
Determining the Volume of Additive X
Next, we need to find the volume of additive X using its density:
Density of X = 1.24 g/ml
Using the formula for density (density = mass/volume), we can rearrange it to find volume:
Volume of X = mass / density = 1240 g / 1.24 g/ml ≈ 1000 ml
Calculating the Volume Ratio
Since we are using 1000 ml of water (1 kg), we can now find the volume ratio of X to water:
Volume ratio of X to water = Volume of X : Volume of water = 1000 ml : 1000 ml = 1:1
Final Volume by Volume Ratio
However, we need to express this in terms of the options provided. The closest option that matches our findings is:
- 1:1 is equivalent to 50:50 when expressed in a percentage format.
Thus, the correct answer is option 3) 50:50.
