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Ch.1 - Matter, Measurement & Problem Solving
Tro - Chemistry: A Molecular Approach 4th Edition
Tro4th EditionChemistry: A Molecular ApproachISBN: 9780134112831당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Tro 4th EditionCh.1 - Matter, Measurement & Problem Solving문제 137
1장, 문제 137

Kinetic energy can be defined as (1/2)mv^2 or as (3/2)PV. Show that the derived SI units of each of these terms are those of energy. (Pressure is force/area and force is mass * acceleration.)

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Start by analyzing the first expression for kinetic energy: \( \frac{1}{2}mv^2 \). Here, \( m \) is mass with SI units of kilograms (kg), and \( v \) is velocity with SI units of meters per second (m/s).
Calculate the units for \( v^2 \): since velocity \( v \) is in m/s, \( v^2 \) will be in \((\text{m/s})^2 = \text{m}^2/\text{s}^2\).
Combine the units for mass and velocity squared: \( \text{kg} \times \text{m}^2/\text{s}^2 = \text{kg} \cdot \text{m}^2/\text{s}^2 \), which are the units of energy, known as a joule (J).
Now, consider the second expression for kinetic energy: \( \frac{3}{2}PV \). Here, \( P \) is pressure with units of force/area, and \( V \) is volume with units of cubic meters (m^3).
Express pressure \( P \) in terms of its fundamental units: force is mass times acceleration (\( \text{kg} \cdot \text{m/s}^2 \)), and area is \( \text{m}^2 \), so pressure \( P \) is \( \text{kg} \cdot \text{m/s}^2/\text{m}^2 = \text{kg}/(\text{m} \cdot \text{s}^2) \). Multiply by volume \( V \) to get \( \text{kg} \cdot \text{m}^2/\text{s}^2 \), which are again the units of energy, a joule (J).

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Kinetic Energy Formula

Kinetic energy (KE) is defined as the energy an object possesses due to its motion, mathematically expressed as KE = (1/2)mv^2, where m is mass and v is velocity. This formula indicates that kinetic energy is directly proportional to the mass of the object and the square of its velocity, highlighting how changes in either variable significantly affect the energy.
추천 영상:
가이드 코스
02:27
Kinetic Energy Formulas

Pressure and Its Units

Pressure (P) is defined as force (F) applied per unit area (A), expressed as P = F/A. The SI unit of pressure is the pascal (Pa), which is equivalent to one newton per square meter (N/m²). Understanding pressure is crucial for deriving energy expressions, as it relates to the force exerted by gas molecules in a given volume.
추천 영상:
가이드 코스
01:15
Pressure Units

Dimensional Analysis

Dimensional analysis is a mathematical technique used to convert units and verify the consistency of equations by ensuring that both sides have the same dimensions. In the context of energy, it involves checking that the derived units from different expressions (like (1/2)mv^2 and (3/2)PV) yield the same unit of energy, which is the joule (J) in the SI system.
추천 영상:
가이드 코스
06:11
Dimensional Analysis
관련 실천
교과서 질문

Mercury is often used in thermometers. The mercury sits in a bulb on the bottom of the thermometer and rises up a thin capillary as the temperature rises. Suppose a mercury thermometer contains 3.380 g of mercury and has a capillary that is 0.200 mm in diameter. How far does the mercury rise in the capillary when the temperature changes from 0.0 °C to 25.0 °C? The density of mercury at these temperatures is 13.596 g/cm3 and 13.534 g/cm3, respectively

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