Joules to calories

1. The problem statement, all variables and given/known data
1 calorie is the energy needed to raise the temperature of 1 gram of water by 1 degree Celcius, i.e., the specific heat of water is 1 calorie/g-°C. How many joules is 1 calorie?

2. Relevant equations
ΔU = nαkΔT

3. The attempt at a solution
I attempted to use equipartition to solve this problem.
α of water should be 6*1/2 = 3.
1 gram of water should have 6.02*10^23 * 1/18 = 3.34 * 10^22 molecules
So to increase the temperature of 1 gram of water by 1° C or K, I obtained 3.34*10^22*3*1.38*10^-23*1 = 1.38 J.

However, it is well established that 4.184 J = 1 calorie. What am I doing wrong?

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