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Suppose that the change in V\(_m\) was caused by the entry of Ca\(^{2+}\) instead of Na\(^+\). How many Ca\(^{2+}\) ions would have to enter the cell per unit membrane to produce the change? (a) Half as many as for Na\(^+\); (b) the same as for Na\(^+\); (c) twice as many as for Na\(^+\); (d) cannot say without knowing the inside and outside concentrations of Ca\(^{2+}\).

Short Answer

Expert verified
(a) Half as many as for Na\(^+\).

Step by step solution

01

Understand the question

We need to determine how the entry of Ca\(^{2+}\) ions affects the change in membrane potential \(V_m\) in comparison to Na\(^+\) ions. Different options are given based on how many Ca\(^{2+}\) ions are needed relative to Na\(^+\).
02

Recall the charge difference

Ca\(^{2+}\) ions have a charge of +2, compared to Na\(^+\) ions which have a charge of +1. This means each Ca\(^{2+}\) ion carries double the charge compared to each Na\(^+\) ion.
03

Determine required ions

Since Ca\(^{2+}\) is doubly charged, the entry of one Ca\(^{2+}\) ion would produce the same charge difference as two Na\(^+\) ions. Therefore, half as many Ca\(^{2+}\) ions are needed to effect the same change in \(V_m\) as Na\(^+\) ions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calcium Ions
Calcium ions, denoted as Ca\(^{2+}\), are positively charged ions that play a crucial role in various cellular processes. These ions carry a positive charge of +2, which means they are doubly charged compared to other common cations like sodium ions. This double charge is significant in the context of membrane potential, as it impacts how these ions influence charge distribution and changes in potential.

In cellular environments, calcium ions are often found in low concentrations inside cells but higher concentrations outside. This gradient is essential for signal transduction and muscle contraction. When calcium ions enter a cell, they can trigger various responses, such as the release of neurotransmitters or the initiation of muscle contraction.
Sodium Ions
Sodium ions, or Na\(^+\), are fundamental to maintaining the cell's resting membrane potential and facilitating action potentials. With a single positive charge, sodium ions are involved in transporting other ions or molecules across cell membranes.

In neurons, the rapid influx of sodium ions during an action potential leads to depolarization, a critical step in the transmission of nerve impulses. Sodium-potassium pumps work to maintain a high concentration of sodium ions outside cells and potassium ions inside, which is vital for various cellular functions.
  • Sodium ions have a +1 charge.
  • Key players in nerve impulse transmission.
  • Integral to maintaining balance in ion concentrations and membrane potential.
Ion Concentration
Ion concentration refers to the amount of specific ions present in and around a cell. This concentration plays a pivotal role in determining the membrane potential, which is essentially the voltage difference across a cell's membrane.

The membrane potential is influenced by the differential distribution of ions like calcium and sodium across the membrane, as quantified by the Nernst equation. This equation considers the intracellular and extracellular concentrations of ions to calculate the equilibrium potential.

Changes in ion concentration can lead to alterations in membrane potential, thereby impacting cellular activities such as muscle contraction, nerve firing, and the opening or closing of ion channels.
Electrochemical Gradient
The electrochemical gradient is a combination of two forces: the chemical gradient and the electrical gradient. It drives the movement of ions across membranes.

The chemical gradient refers to the difference in ion concentration between the inside and outside of the cell, whereas the electrical gradient is the difference in charge across the membrane.
  • Driving force for ion movement.
  • Maintains crucial cellular activities.
  • Essential for establishing membrane potential.

These gradients are at the heart of many physiological processes, including the transmission of nerve impulses and the operation of cardiac muscles. By maintaining a balance between these gradients, cells can ensure proper function and responsiveness to external stimuli.

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Most popular questions from this chapter

A cylindrical capacitor has an inner conductor of radius 2.2 mm and an outer conductor of radius 3.5 mm. The two conductors are separated by vacuum, and the entire capacitor is 2.8 m long. (a) What is the capacitance per unit length? (b) The potential of the inner conductor is 350 mV higher than that of the outer conductor. Find the charge (magnitude and sign) on both conductors.

A parallel-plate air capacitor is made by using two plates 12 cm square, spaced 3.7 mm apart. It is connected to a 12-V battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field between the plates? (d) What is the energy stored in the capacitor? (e) If the battery is disconnected and then the plates are pulled apart to a separation of 7.4 mm, what are the answers to parts (a)-(d)?

A 12.5-\(\mu\)F capacitor is connected to a power supply that keeps a constant potential difference of 24.0 V across the plates. A piece of material having a dielectric constant of 3.75 is placed between the plates, completely filling the space between them. (a) How much energy is stored in the capacitor before and after the dielectric is inserted? (b) By how much did the energy change during the insertion? Did it increase or decrease?

A capacitor is made from two hollow, coaxial, iron cylinders, one inside the other. The inner cylinder is negatively charged and the outer is positively charged; the magnitude of the charge on each is 10.0 pC. The inner cylinder has radius 0.50 mm, the outer one has radius 5.00 mm, and the length of each cylinder is 18.0 cm. (a) What is the capacitance? (b) What applied potential difference is necessary to produce these charges on the cylinders?

A parallel-plate capacitor with only air between the plates is charged by connecting it to a battery. The capacitor is then disconnected from the battery, without any of the charge leaving the plates. (a) A voltmeter reads 45.0 V when placed across the capacitor. When a dielectric is inserted between the plates, completely filling the space, the voltmeter reads 11.5 V. What is the dielectric constant of this material? (b) What will the voltmeter read if the dielectric is now pulled partway out so it fills only one-third of the space between the plates?

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