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For 1 mole of an ideal gas, the molar specific heat capacities at constant pressure ($C_p$) and constant volume ($C_v$) satisfy which relation?
The NCERT text explicitly states: 'For an ideal gas, the molar specific heat capacities at constant pressure and volume satisfy the relation $C_p - C_v = R$' where R is the universal gas constant.
If equal amounts of heat are added to equal masses of two different substances, A and B, and substance A experiences a smaller temperature change than substance B, what can be concluded about their specific heat capacities?
Given $\Delta Q = ms \Delta T$. If $\Delta Q$ and $m$ are constant for both substances, then $s \propto 1/\Delta T$. A smaller temperature change ($\Delta T$) implies a larger specific heat capacity ($s$). Therefore, $s_A > s_B$.
Why is the term 'mechanical equivalent of heat' now considered superfluous in SI units?
The NCERT text states: 'Since in SI units, we use the unit joule for heat, work or any other form of energy, the term mechanical equivalent is now superfluous and need not be used.' It was essentially a conversion factor between calories and joules.
Considering the provided table for specific heat capacities, which solid requires the most heat to increase the temperature of 1 kg of it by 1 K?
Referring to Table 10.3 (Specific heat capacity of some substances), Aluminium has a specific heat capacity of 900.0 J kg$^{-1}$ K$^{-1}$, Copper 386.4 J kg$^{-1}$ K$^{-1}$, Iron 450 J kg$^{-1}$ K$^{-1}$, and Lead 127.7 J kg$^{-1}$ K$^{-1}$. The substance with the highest specific heat capacity requires the most heat for a given temperature change and mass. Aluminium has the highest value among the given options.
What is the relationship between the heat capacity (S) of a substance and its specific heat capacity (s)?
From the NCERT text, specific heat capacity is defined as $s = \frac{\Delta Q}{m\Delta T}$. Heat capacity (S) is defined as $S = \frac{\Delta Q}{\Delta T}$. Therefore, by multiplying $s$ by $m$, we get $ms = m \frac{\Delta Q}{m\Delta T} = \frac{\Delta Q}{\Delta T} = S$. So, $S = ms$.
The molar specific heat capacity for solids (like in Table 11.1) is generally found to be around 3R. What does this prediction based on the law of equipartition of energy suggest about the energy distribution within the solid?
The NCERT text clearly states: 'Consider a solid of N atoms, each vibrating about its mean position. An oscillator in one dimension has average energy of $2 \times \frac{1}{2} k_B T = k_B T$. In three dimensions, the average energy is $3 k_B T$.' For a mole, this leads to $U = 3 RT$ and thus $C = 3R$. Each degree of freedom of a 3D harmonic oscillator (vibration) contributes $\frac{1}{2}k_BT$ for kinetic and $\frac{1}{2}k_BT$ for potential energy, totaling $k_BT$ per dimension, and thus $3k_BT$ for 3 dimensions per atom.
A 0.047 kg aluminium sphere at 100 °C is transferred to a copper calorimeter containing water. If the final steady state temperature is 23 °C, and the specific heat capacity of aluminium ($s_{Al}$) is to be calculated by equating heat lost by aluminium to heat gained by water and calorimeter, which of the following is true for the heat lost by aluminium?
Heat lost or gained is given by $Q = ms\Delta T$. The mass of the aluminium sphere ($m_1$) is 0.047 kg, the specific heat capacity is $s_{Al}$, and the change in temperature ($\Delta T$) for the aluminium sphere is from 100 °C down to 23 °C, so $\Delta T = (100 - 23)$ °C. Thus, $Q_{lost}$ by aluminium $= m_1 s_{Al} (T_{initial} - T_{final}) = 0.047 \times s_{Al} \times (100 - 23)$.
Which of the following statements about a unit vector is INCORRECT?
As per the NCERT text, 'A unit vector is a vector of unit magnitude and points in a particular direction. It has no dimension and unit. It is used to specify a direction only.' If a unit vector $\hat{n}$ is multiplied by a positive scalar $\lambda$, the result is $\lambda\hat{n}$. While the magnitude changes to $\lambda$, the direction remains the same as the original unit vector. So the unit vector itself, representing direction, does not change, only the resulting vector does. Statement 4 is incorrect because the unit vector itself (its direction) does not change when multiplied by a scalar, only the magnitude of the resulting vector changes.
A vector $\vec{A}$ lies in the x-y plane. If its x-component is $A_x$ and y-component is $A_y$, then $\vec{A}$ can be expressed as:
According to equation (3.12) in the NCERT text, 'Thus, $\vec{A} = A_x \hat{i} + A_y \hat{j}$'. This represents the resolution of a vector into its component vectors along the unit vectors $\hat{i}$ (x-axis) and $\hat{j}$ (y-axis).
If $\vec{A}$ is a vector and $\hat{n}$ is a unit vector along the direction of $\vec{A}$, which of the following relations is correct?
From the summary point 9 in the NCERT text, 'A unit vector associated with a vector A has magnitude 1 and is along the vector A: $\hat{n} = \frac{\vec{A}}{|A|}$'.
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