characteristic time constant

Examples of characteristic time constant in the following topics:

• Constant Velocity

  • An object moving with constant velocity must have a constant speed in a constant direction.
  • If an object is moving at constant velocity, the graph of distance vs. time ($x$ vs.
  • $t$) shows the same change in position over each interval of time.
  • You can also obtain an object's velocity if you know its trace over time.
  • When an object is moving with constant velocity, it does not change direction nor speed and therefore is represented as a straight line when graphed as distance over time.

• Motion with Constant Acceleration

  • Constant acceleration occurs when an object's velocity changes by an equal amount in every equal time period.
  • An object experiencing constant acceleration has a velocity that increases or decreases by an equal amount for any constant period of time.
  • It is defined as the first time derivative of velocity (so the second derivative of position with respect to time):
  • When it is not, we can either consider it in separate parts of constant acceleration or use an average acceleration over a period of time.
  • Due to the algebraic properties of constant acceleration, there are kinematic equations that relate displacement, initial velocity, final velocity, acceleration, and time.

• Equilibrium Constant Expression

  • The equilibrium constant is an expression that gives the ratio of reactants and products at equilibrium.
  • This constant is known as the equilibrium constant.
  • Their activity is 1, so they do not need to be written in the equilibrium constant.
  • At time = 0, the rate of the forward reaction is high and the rate of the reverse reaction is low.
  • The equilibrium constant,denoted by K, is the ratio of products to reactants at equilibrium.

• Planck's Quantum Hypothesis and Black Body Radiation

  • A black body in thermal equilibrium (i.e. at a constant temperature) emits electromagnetic radiation called black body radiation.
  • Black body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature.
  • where $B$ is the spectral radiance of the surface of the black body, $T$ is its absolute temperature, $\lambda$ is wavelength of the radiation, $k_B$ is the Boltzmann constant, $h$ is the Planck constant, and $c$ is the speed of light.
  • It is not a surprise that he introduced Planck constant $h = 6.626 \times 10^{-34} J \cdot s$ for the first time in his derivation of the Planck's law.

• Acid Dissociation Constant (Ka)

  • The acid dissociation constant (Ka) is the measure of the strength of an acid in solution.
  • The acid dissociation constant (Ka) is a quantitative measure of the strength of an acid in solution.
  • In the above reaction, HA (the generic acid), A- (the conjugate base of the acid), and H+ (the hydrogen ion or proton) are said to be in equilibrium when their concentrations do not change over time.
  • The logarithmic constant (pKa) is equal to -log10(Ka).
  • Acetic acid is a weak acid with an acid dissociation constant $K_a=1.8\times 10^{-5}$ .

• Simple harmonic oscillation

  • The constant of proportionality is called the spring constant and is usually denoted by k.
  • So this is its natural or characteristic frequency.
  • Let's continue to refer to this characteristic frequency as $\omega_0$ to emphasize the fact that it is a constant for a given spring/mass system.
  • So $B$ must equal whatever velocity the mass has when it zips through the origin, divided by the characteristic frequency $\omega_0$.
  • Since the $t$ disappears, we see that the energy is constant with time, and thus energy is conserved.

• Renal Clearance

  • In renal physiology, clearance is a measurement of the renal excretion ability, which measures the amount of plasma from which a substance is removed from the body over an interval of time.
  • Each substance has its own specific clearance that depends on its unique filtration characteristics.
  • Clearance can be either a constant or variable component over time, depending on the type of substance.
  • Additionally, the characteristics of the substance of interest will also determine some components of clearance.
  • These types of clearance all add up to a summation known as total body clearance, which refers to the removal of a substance from the plasma over time, incorporating all routes of removal in the body.

• Specific Heat for an Ideal Gas at Constant Pressure and Volume

  • An ideal gas has different specific heat capacities under constant volume or constant pressure conditions.
  • where the partial derivatives are taken at: constant volume and constant number of particles, and at constant pressure and constant number of particles, respectively.
  • The heat capacity ratio or adiabatic index is the ratio of the heat capacity at constant pressure to heat capacity at constant volume.
  • It is a simple equation relating the heat capacities under constant temperature and under constant pressure.
  • In addition, molecules in the gas may pick up many characteristic internal vibrations.

• Angular vs. Linear Quantities

  • Imagine a particle moving in a circle around a point at a constant speed.
  • At any instant in time, the particle is moving in a particular straight-line direction with that speed.
  • Thus, while the object moves in a circle at constant speed, it undergoes constant linear acceleration to keep it moving in a circle.
  • However, it's angular velocity is constant since it continually sweeps out a constant arc length per unit time.
  • Constant angular velocity in a circle is known as uniform circular motion.

• Different Types of Currents

  • If the sources are constant (DC) sources, the result is a DC circuit.
  • A direct current circuit is an electrical circuit that consists of any combination of constant voltage sources, constant current sources, and resistors.
  • In this case, the circuit voltages and currents are independent of time.
  • The solution to these equations usually contain a time varying or transient part as well as constant or steady state part.
  • The two Kirchoff laws along with the current-voltage characteristic (I-V curve) of each electrical element completely describe a circuit.