According to the provided reference, salt (sodium chloride, NaCl) decomposes in water into sodium (Na+) and chloride (Cl-) ions. This process, within the context of the reference, is described as following the law of uninhibited decay.
Understanding Salt Decomposition (According to the Reference)
When table salt (NaCl) is placed in water, it undergoes a process called dissolution. The reference specifically uses the term "decomposes" to describe this process, where the ionic bonds holding the sodium and chloride together break, allowing the individual ions to separate and disperse throughout the water.
- Chemical Equation: NaCl(s) → Na+(aq) + Cl-(aq)
- What Happens: Solid salt (NaCl) breaks down into positively charged sodium ions (Na+) and negatively charged chloride ions (Cl-) when dissolved in water.
- Key Environment: This "decomposition" as described in the reference occurs specifically in water.
The reference further models the rate at which the initial amount of solid salt decreases over time using the law of uninhibited decay. This mathematical model is often applied to processes where the rate of decrease of a substance is proportional to the amount of the substance present.
The Law of Uninhibited Decay Model
The reference states that the decomposition follows the law of uninhibited decay. This law is typically represented by the formula:
A(t) = A₀ * e^(-kt)
Where:
A(t)is the amount of the substance remaining at timet.A₀is the initial amount of the substance.eis the base of the natural logarithm (approximately 2.71828).kis the decay constant (a positive value determining the rate of decay).tis the time elapsed.
The reference provides specific data points related to this decay process to illustrate how the amount of solid salt changes over time:
- Initial Amount (A₀): 25 kilograms
- Amount after 10 hours (A(10)): 15 kilograms
Using this information, we can determine the decay constant k.
Calculating the Decay Constant (k)
- Start with the decay formula:
A(t) = A₀ * e^(-kt) - Substitute the known values:
15 = 25 * e^(-k * 10) - Divide both sides by 25:
15 / 25 = e^(-10k) - Simplify the fraction:
0.6 = e^(-10k) - Take the natural logarithm (ln) of both sides:
ln(0.6) = ln(e^(-10k)) - Use the logarithm property
ln(e^x) = x:ln(0.6) = -10k - Solve for
k:k = ln(0.6) / -10
Using a calculator, ln(0.6) is approximately -0.5108.
k ≈ -0.5108 / -10
k ≈ 0.05108 per hour
So, the decay constant for this specific scenario is approximately 0.05108 per hour. This means that each hour, the amount of remaining salt decreases by about 5.108% of the amount present at the start of that hour, according to this model.
Applying the Decay Model
The reference poses two questions based on this model and the calculated decay constant:
- How much salt is left after 1 day?
- How long does it take until 1/2 kilogram of salt is left?
Let's solve these problems to fully incorporate the information from the reference.
Amount of Salt Left After 1 Day
1 day is equal to 24 hours. We use the decay formula with A₀ = 25, k ≈ 0.05108, and t = 24.
A(24) = 25 * e^(-0.05108 * 24)
A(24) = 25 * e^(-1.22592)
Calculate e^(-1.22592): This is approximately 0.2935.
A(24) ≈ 25 * 0.2935
A(24) ≈ 7.3375 kilograms
So, according to the law of uninhibited decay model described in the reference, approximately 7.34 kilograms of salt is left after 1 day.
Here is a summary table:
| Time (hours) | Calculated Amount Remaining (kg) |
|---|---|
| 0 | 25 (Initial) |
| 10 | 15 (Given) |
| 24 (1 day) | ≈ 7.34 |
Time Until 1/2 Kilogram is Left
We want to find the time t when A(t) = 0.5 kg. We use the decay formula with A₀ = 25, k ≈ 0.05108, and A(t) = 0.5.
0.5 = 25 * e^(-0.05108 * t)
- Divide both sides by 25:
0.5 / 25 = e^(-0.05108 * t) - Simplify the fraction:
0.02 = e^(-0.05108 * t) - Take the natural logarithm of both sides:
ln(0.02) = ln(e^(-0.05108 * t)) - Use the logarithm property
ln(e^x) = x:ln(0.02) = -0.05108 * t
Calculate ln(0.02): This is approximately -3.912.
-3.912 ≈ -0.05108 * t
- Solve for
t:t ≈ -3.912 / -0.05108
t ≈ 76.58hours
Thus, according to the model, it takes approximately 76.58 hours until 1/2 kilogram of salt is left.
To put 76.58 hours into context:
- 76 hours is 3 days and 4 hours.
- 76.58 hours is approximately 3 days, 4 hours, and 35 minutes.
In summary, based on the provided reference, salt decomposes in water into sodium and chloride ions, and this process's rate is described using the law of uninhibited decay. The decay model parameters derived from the reference's data predict specific amounts remaining and times to reach certain amounts.
