$$N_t = N_0 * (1/2)^{t/t_{1/2}}$$
Where N_t is the amount of substance at time t, N_0 is the initial amount of substance, t is the time elapsed and t_{1/2} is the half-life of the substance.
Given:
N_t = 2.0 grams
N_0 = 32 grams
t_{1/2} = 17 days
Substituting these values into the formula:
$$2.0 = 32 * (1/2)^{t/17}$$
Dividing both sides by 32:
$$\frac{2.0}{32} = (1/2)^{t/17}$$
Simplifying:
$$0.0625 = (1/2)^{t/17}$$
Taking the logarithm of both sides:
$$\log(0.0625) = \frac{t}{17} * \log(1/2)$$
Solving for t:
$$t = \frac{17 \times \log(0.0625)}{\log(1/2)}$$
$$t \approx 51 days$$
Therefore, it takes approximately 51 days for 32 grams of palladium-103 to decay to 2.0 grams.
