Question

The mass of substance is \(75.5 \mathrm{gm}\) and its volume is \(25 \mathrm{~cm}^{3}\). It's density up to the correct significant figure is \(\ldots \ldots \ldots \ldots \ldots\) (a) \(3.02 \mathrm{gm} / \mathrm{cm}^{3}\) (b) \(3.200 \mathrm{gm} / \mathrm{cm}^{3}\) (c) \(3.02 \mathrm{gm} / \mathrm{cm}^{3}\) (d) \(3.1 \mathrm{gm} / \mathrm{cm}^{3}\)

Short answer

The correct answer is options (a) and (c): \( 3.02 \ \mathrm{gm/cm^{3}} \).

Step-by-step solution

Identify the given values

The mass of the substance is given as \(75.5 \ \mathrm{gm}\), and the volume is \(25 \ \mathrm{cm^{3}}\).

Calculate the density

To calculate the density, we will use the formula Density = Mass / Volume given by: Density = \( \frac{75.5 \ \mathrm{gm}}{25 \ \mathrm{cm^{3}}} \)

Simplify the expression

We can simplify the expression by dividing 75.5 by 25: Density = \( \frac{75.5}{25} \ \mathrm{gm/cm^{3}} \) = \( 3.02 \ \mathrm{gm/cm^{3}} \)

Write the answer in the correct significant figure format

The answer obtained is already in the correct significant figure format (two decimal places), as both the mass (75.5 gm) and volume (25 cm³) have two significant figures. So the answer is: Density = \( 3.02 \ \mathrm{gm/cm^{3}} \) Comparing the calculated density with the options provided: (a) \(3.02 \ \mathrm{gm/cm^{3}}\) - Correct (b) \(3.200 \ \mathrm{gm/cm^{3}}\) - Incorrect; too many significant figures (c) \(3.02 \ \mathrm{gm/cm^{3}}\) - Repeated option, still correct (d) \(3.1 \ \mathrm{gm/cm^{3}}\) - Incorrect; incorrect significant figure The correct answer is options (a) and (c): \( 3.02 \ \mathrm{gm/cm^{3}} \).

Key concepts

Significant Figures

Significant figures are vital in scientific measurements because they convey the precision of a measurement. These figures include all known digits plus one estimated digit, helping to indicate the certainty of your measurements. When looking at a value such as 75.5, the significant figures tell us that we have three digits of known information. This plays a crucial role in calculations.

• Why are they important? They help ensure that results are not overrepresented with false precision.
• When dividing or multiplying, the result should have the same number of significant figures as the number with the least significant figures used in the calculation.

In our density calculation, the mass has three significant figures, while the volume has two. The answer, therefore, needs to be rounded to the lesser quantity, which is two significant figures.

Mass

Mass is the quantity of matter in a substance, typically measured in grams or kilograms. It serves as one of the essential elements in various physics and chemistry calculations, such as density.

• In the exercise, the substance's mass was given as 75.5 grams.
• This measurement provides a vital piece of information needed to calculate density.

The precision of this value affects how you present your final answer when performing calculations. As it has three significant figures, it implies a fairly accurate measurement was taken.

Volume

Volume measures the space that a substance occupies and is often measured in cubic centimeters (\(cm^3\)), liters, or milliliters. Understanding how to measure and interpret volume is crucial for comprehending the concept of density.

• The provided volume in the exercise was 25 \(cm^3\).
• A key point here is that the number of significant figures in the volume, two, informed the density calculation.

When combined with mass, volume allows us to determine the density, which tells us how compact or concentrated the substance is. Accurately knowing the volume helps in these calculations and affects how your results, especially with significant figures, should be reported.