Step 1: Recall the half-life link. The half-life is the time for half the substance to decay. It connects to the decay constant by \[ t_{1/2} = \frac{\ln 2}{\lambda} \]
Step 2: Rearrange for the decay constant. Solve the formula for $\lambda$. \[ \lambda = \frac{\ln 2}{t_{1/2}} \]
Step 3: Put in the half-life. The half-life is $10$ days. \[ \lambda = \frac{\ln 2}{10} \]
Step 4: Use the value of $\ln 2$. We know $\ln 2 \approx 0.693$. \[ \lambda = \frac{0.693}{10} \]
Step 5: Do the division. \[ \lambda = 0.0693 \text{ per day} \]
Step 6: State the answer. The decay constant is about $0.0693$ per day. \[ \boxed{\lambda = 0.0693 \text{ /day}} \]