Scientific Study: Variables, Units, and Principle of Homogeneity

Scientific Study: Foundations and Reproducibility

• Definition: Scientific study is the systematic learning of any event, process, material, or transformation. This process is grounded in established theories, models, and experimentation.

• Key Characteristics: The most critical aspects of a scientific study are its systematic nature, reproducibility, and the repeatability of results.

• Reproducibility: This is defined as the recurrence of identical results with a high degree of reliability through experiment, observation, or data analysis when the study is conducted under the exact same set of conditions.

• Illustrative Case Study: Hydrogen Gas Preparation

  • Goal: Preparation of hydrogen gas through the reaction of magnesium ($Mg$) metal and acidulated water.

  • Experimental Conditions:

    • Temperature: $25\,^\circ\text{C}$

    • Pressure: $1\,\text{atm}$

    • Mass of Magnesium ($Mg$): $12\,\text{mg}$

    • Volume of acidulated water: $30\,\text{mL}$

  • Result: The volume of hydrogen gas liberated is found to be $22.4\,\text{cc}$.

  • Repeatability Demonstration: If Student A, Student B, and Student C all perform the experiment under these specific conditions, they should all produce exactly $22.4\,\text{cc}$ of hydrogen gas.

• Advancement in Research: Modification, clarification, verification, and advancement of results in any experiment are achieved by systematically changing conditions or variables.

Variables in Scientific Research

• Definition: Variables refer to any factors, traits, or parameters that can be controlled, changed (varied), or measured relative to another parameter during an experiment. Examples include temperature, pressure, volume, and mass.

• Experimental Event 1: Changing Acceleration (Constant Mass)

  • Setup: An iron ball with a mass of $1\,\text{kg}$ is thrown with three different levels of acceleration.

  • Data Table:         | Mass of ball | Acceleration ($\text{m/s}^2$) | Force ($F = m \times a$) | Remarks |         | :--- | :--- | :--- | :--- |         | $1\,\text{kg}$ | $1$ | $1\,\text{N}$ | - |         | $1\,\text{kg}$ | $2$ | $2\,\text{N}$ | - |         | $1\,\text{kg}$ | $3$ | $3\,\text{N}$ | Highest force; more impact |

  • Analysis: The variable "mass" is held constant, while "acceleration" is changed. The force depends directly on the acceleration.

• Experimental Event 2: Changing Mass (Constant Acceleration)

  • Setup: Three iron balls of different masses ($1\,\text{kg}$, $2\,\text{kg}$, and $3\,\text{kg}$) are moved with the same acceleration of $3\,\text{m/s}^2$.

  • Data Table:         | Mass of ball | Acceleration ($\text{m/s}^2$) | Force ($F = m \times a$) | Remarks |         | :--- | :--- | :--- | :--- |         | $1\,\text{kg}$ | $3$ | $3\,\text{N}$ | - |         | $2\,\text{kg}$ | $3$ | $6\,\text{N}$ | - |         | $3\,\text{kg}$ | $3$ | $9\,\text{N}$ | Highest force; more impact |

  • Analysis: The variable "acceleration" is held constant, while "mass" is changed. The force depends directly on the mass.

Categorization of Scientific Variables

• 1. Independent Variable:

  • A variable that is fixed and not affected by other variables during the experiment.

  • In scientific research, this is the factor that is purposefully changed or manipulated to observe its effect.

• 2. Dependent Variable:

  • The variable that is being tested and measured. It changes in response to manipulations of the independent variable.

• 3. Controlled Variable:

  • Those variables kept constant throughout the entire course of experimentation.

  • Keeping these variables constant ensures that the observed effect on the dependent variable is caused solely by the independent variable.

  • Example: In a study of salt solubility in water at different temperatures:

    • Independent Variable: Temperature.

    • Dependent Variable: Solubility of salt.

    • Controlled Variables: Pressure, volume of water, and type of container used.

• 4. Extraneous Variable:

  • Often called "extra" variables, these have less impact and are often regarded as less important.

  • However, they can introduce errors into the results. Awareness of these variables allows for higher accuracy.

  • Example: Humidity levels during a study of salt solubility.

Principles of Variable Control and Representation

• Management Rules for Research:

  1. An experiment should have strictly one independent variable.

  2. A single dependent variable is required for an experiment.

  3. All other variables except the independent and dependent ones must be controlled.

• Importance of Controlling Variables:

  • Establishes causal relationships between variables of interest.

  • Avoids research bias.

  • Small variations in research variables can strongly affect outcomes; control prevents confusion by focusing on a specific factor.

• Mathematical Expression:

  • Relationships are expressed as equations. The dependent variable is written on the left side, and the independent variable is written on the right side.

  • Example: $F = m \times a$

    • If $a$ is the independent variable and $m$ is controlled, $F$ is the dependent variable.

    • If $m$ is the independent variable and $a$ is controlled, $F$ is the dependent variable.

• Graphical Representation:

  • X-axis: The independent variable (e.g., Heat).

  • Y-axis: The dependent variable (e.g., Temperature).

• Illustration (Gas Laws Context):

  • Statement: For a certain mass of gas, volume is inversely proportional to pressure at constant temperature.

  • Controlled Variables: Mass and Temperature.

  • Independent Variable: Pressure.

  • Dependent Variable: Volume.

Units of Measurement: Fundamental and Derived

• Definition of a Unit: A standard measure used to express a measured amount.

• 1. Fundamental Units:

  • Also known as basic units, these are independent and cannot be expressed in terms of any other units.

  • There are seven fundamental units and two supplementary units.

  • The Seven Fundamental Units:

    1. Length: Meter ($\text{m}$)

    2. Mass: Kilogram ($\text{kg}$)

    3. Time: Second ($\text{s}$)

    4. Temperature: Kelvin ($\text{K}$)

    5. Electric current: Ampere ($\text{A}$)

    6. Luminous intensity: Candela ($\text{cd}$)

    7. Amount of a substance: Mole ($\text{mol}$)

  • The Two Supplementary Units:

    1. Plane angle: Radian ($\text{rad}$)

    2. Solid angle: Steradian ($\text{sr}$)

• 2. Derived Units:

  • These units are dependent on fundamental units and are obtained via algebraic operations (multiplication or division).

  • Example: Area = length $\times$ length = $\text{m} \times \text{m} = \text{m}^2$.

Comparison: Fundamental vs. Derived Units

Detailed Table of Derived Units and Dimensions

Principle of Homogeneity of Equation

• Definition: This principle states that the dimensions (units) of each term in a dimensional equation must be the same on both sides of the equation.

• Application: It is used to perform unit-wise analysis to check the validity or "truthiness" of a scientific equation.

• Verification Example: Force ($F = mg$):

  • Left Hand Side (LHS): Force ($F$) is measured in Newtons ($\text{N}$). As established in derived units: $\text{N} = \text{kg} \cdot \text{m} \cdot \text{s}^{-2}$.

  • Right Hand Side (RHS): Mass ($m$) $\times$ acceleration due to gravity ($g$).

    • Unit of mass ($m$) = $\text{kg}$

    • Unit of acceleration due to gravity ($g$) = $\text{m} \cdot \text{s}^{-2}$

    • RHS Product = $\text{kg} \cdot \text{m} \cdot \text{s}^{-2}$

  • Conclusion: Since LHS ($\text{kg} \cdot \text{m} \cdot \text{s}^{-2}$) = RHS ($\text{kg} \cdot \text{m} \cdot \text{s}^{-2}$), the relation is correct.