Question

A \(37.2-\mathrm{g}\) sample of lead \((\mathrm{Pb})\) pellets at \(20^{\circ} \mathrm{C}\) is mixed with a 62.7 -g sample of lead pellets at the same temperature. What are the final mass, temperature, and density of the combined sample? The density of \(\mathrm{Pb}\) at \(20^{\circ} \mathrm{C}\) is \(11.35 \mathrm{~g} / \mathrm{cm}^{3}\).

Short answer

Final mass: 99.9 g. Final temperature: 20°C. Density: 11.35 g/cm³.

Step-by-step solution

Calculate Total Mass

The final mass is simply the sum of the masses of the two samples of lead. We add the mass of the first sample (37.2 g) to the mass of the second sample (62.7 g), so: \[ 37.2 ext{ g} + 62.7 ext{ g} = 99.9 ext{ g} \] Thus, the final mass of the combined lead sample is 99.9 g.

Determine Final Temperature

Since both samples are lead and have the same initial temperature (20°C), when they are mixed together, the final temperature will remain the same because there is no exchange of heat between different temperature bodies. Therefore, the final temperature is \(20^{ ext{o}} ext{C}\).

Calculate Final Volume Using Density

To find the volume of the combined lead sample, we use the formula: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]Given the combined mass is 99.9 g and the density of lead at 20°C is 11.35 g/cm³: \[ \text{Volume} = \frac{99.9 \text{ g}}{11.35 \text{ g/cm}^3} \approx 8.80 \text{ cm}^3 \]

Verify Density Remains Constant

Since the density of a homogeneous substance like lead does not change when masses are combined (assuming temperature remains constant), the density of the combined sample will still be 11.35 g/cm³.

Key concepts

Lead Properties

Lead, represented by the chemical symbol Pb, is a dense metal known for its bluish-white color when freshly cut. One of its prominent properties is high density, making it useful for roles that require heavy weight and mass, such as in weight belts or radiation shielding. It has a relatively low melting point of about 327.5 °C, which means it can be melted and recast efficiently. Due to its softness, lead is also malleable, allowing it to be shaped into sheets or various forms without cracking. Additionally, lead is resistant to corrosion, which means it does not rust like iron and can last a long time in various environments. This resistance is due to the formation of a protective oxide layer that helps to protect it from oxidation and other chemical reactions. With these properties, lead is widely utilized in batteries, construction, and shielding against x-rays and gamma rays.

Temperature Consistency

The exercise shows that when lead samples are combined at the same temperature, the final temperature remains unchanged. This concept demonstrates how a consistent temperature among identical masses leads to no net heat transfer. In other words, heat exchanges only happen between bodies at different temperatures.
Therefore, when two samples of lead at 20°C are mixed, each having the same thermal energy level due to their temperature, there is no need for heat to shift from one sample to the other.
This behavior aligns with the principle of thermal equilibrium, where substances of the same material and consistent temperature experience no internal temperature change when combined. This principle is particularly crucial in experiments and processes that require precise temperature control.

Volume Calculation

The calculation of the volume of a substance involves understanding the relationship between mass and density. Volume is derived from the formula: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \] The exercise solution uses this formula to find the volume of the lead samples when combined. Since mass is given as 99.9 g and density as 11.35 g/cm³, the calculation becomes fairly straightforward: \[ \text{Volume} = \frac{99.9 \text{ g}}{11.35 \text{ g/cm}^3} \approx 8.80 \text{ cm}^3 \]
Volume is a measure of the space occupied by an object, and in this context, it's essential to make sure the unit of mass and the unit used for density correspond to obtain the correct volume.
Understanding how to manipulate the formula is vital for questions of this nature, and practicing with different densities and masses helps grasp the concept more thoroughly.

Density Stability

Density is a significant property of materials, and in this exercise, we see that it remains stable. Density is defined as mass per unit volume: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] For a homogeneous material such as lead, combining different masses at a constant temperature does not alter its density. This means that even when 37.2 g and 62.7 g samples of lead are merged, their collective density stays at 11.35 g/cm³.

• This stability is crucial in practical applications where precise mass-volume relationships are necessary for calculations.
• Knowing that density remains constant allows us to predict the behavior of materials in construction, manufacturing, and scientific industries where specific weight distributions are critical.

Density remains consistent as long as the material remains pure and the temperature stable, ensuring that calculations based on these assumptions hold true in real-world applications.