The nucleus of a uranium atom has a diameter of 1.5×10−14 m and a mass of 4.0×10−25 kg . What is the density of the nucleus?

Ch 18: A Macroscopic Description of Matter

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 18: A Macroscopic Description of Matter

Problem 4

Chapter 18, Problem 4

The nucleus of a uranium atom has a diameter of 1.5×10⁻¹⁴ m and a mass of 4.0×10⁻²⁵ kg . What is the density of the nucleus?

Verified step by step guidance

Step 1: Recall the formula for density, which is given by \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Here, the mass of the nucleus is provided as \( 4.0 \times 10^{-25} \; \text{kg} \).

Step 2: The nucleus is approximately spherical, so the volume of a sphere can be calculated using the formula \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere. The diameter of the nucleus is given as \( 1.5 \times 10^{-14} \; \text{m} \), so the radius \( r \) is half of the diameter: \( r = \frac{1.5 \times 10^{-14}}{2} \; \text{m} \).

Step 3: Substitute the radius \( r \) into the volume formula \( V = \frac{4}{3} \pi r^3 \) to calculate the volume of the nucleus. Ensure that the radius is cubed and multiplied by \( \pi \) and \( \frac{4}{3} \).

Step 4: Substitute the mass \( 4.0 \times 10^{-25} \; \text{kg} \) and the calculated volume into the density formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).

Step 5: Simplify the expression to find the density of the nucleus. Ensure that the units are consistent, and the final density is expressed in \( \text{kg/m}^3 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Density

Density is defined as mass per unit volume, typically expressed in kilograms per cubic meter (kg/m³). It provides a measure of how much matter is contained in a given volume. To calculate density, one can use the formula: density = mass/volume. Understanding density is crucial for analyzing the physical properties of materials, including atomic nuclei.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. In the context of the uranium nucleus, knowing its diameter allows us to find the radius, which is essential for determining the volume. This concept is fundamental in physics and engineering, as it applies to various spherical objects in nature.

Atomic Structure

Atomic structure refers to the arrangement of protons, neutrons, and electrons within an atom. The nucleus, which contains protons and neutrons, is a dense core that accounts for most of the atom's mass. Understanding atomic structure is vital for comprehending the properties of elements, including their density and behavior in chemical reactions.